From 00464bace532f7c893d82a69f366d0b7689f5665 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fran=C3=A7ois=20Pelletier?= <francois@francoispelletier.org>
Date: Thu, 2 Jan 2020 17:11:50 -0500
Subject: [PATCH] first commit

---
 .gitignore                            |   12 +
 LICENSE                               |  204 ++++
 README                                |   13 +
 project.clj                           |    7 +
 src/actuarial/basic.clj               |   24 +
 src/actuarial/core.clj                |    5 +
 src/actuarial/distributions.clj       | 1270 +++++++++++++++++++++++++
 src/actuarial/http.clj                |   50 +
 src/actuarial/measures.clj            |    6 +
 src/actuarial/mixed_distributions.clj |   32 +
 test/actuarial/test/core.clj          |  123 +++
 11 files changed, 1746 insertions(+)
 create mode 100644 .gitignore
 create mode 100644 LICENSE
 create mode 100644 README
 create mode 100644 project.clj
 create mode 100644 src/actuarial/basic.clj
 create mode 100644 src/actuarial/core.clj
 create mode 100644 src/actuarial/distributions.clj
 create mode 100644 src/actuarial/http.clj
 create mode 100644 src/actuarial/measures.clj
 create mode 100644 src/actuarial/mixed_distributions.clj
 create mode 100644 test/actuarial/test/core.clj

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..f10a7aa
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,12 @@
+pom.xml
+pom.xml.asc
+*jar
+/lib/
+/classes/
+/target/
+/checkouts/
+.lein-deps-sum
+.lein-repl-history
+.lein-plugins/
+.lein-failures
+/bin/
diff --git a/LICENSE b/LICENSE
new file mode 100644
index 0000000..3260e4f
--- /dev/null
+++ b/LICENSE
@@ -0,0 +1,204 @@
+Eclipse Public License - v 1.0
+
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diff --git a/README b/README
new file mode 100644
index 0000000..9512e80
--- /dev/null
+++ b/README
@@ -0,0 +1,13 @@
+# actuarial
+
+Probability distributions for actuarial models
+
+## Usage
+
+Use as a function library
+
+## License
+
+Copyright (C) 2014 François Pelletier
+
+Distributed under the Eclipse Public License, the same as Clojure.
diff --git a/project.clj b/project.clj
new file mode 100644
index 0000000..9845065
--- /dev/null
+++ b/project.clj
@@ -0,0 +1,7 @@
+(defproject actuarial "1.0.0-SNAPSHOT"
+  :description "Actuarial functions for Clojure using Incanter and Weka"
+  :dependencies [[org.clojure/clojure "1.6.0"]
+                 [weka "3.6.2"]
+                 [incanter/incanter-core "1.4.1"]
+                 [ring "1.3.2"]
+                 [org.clojure/data.xml "0.0.8"]])
\ No newline at end of file
diff --git a/src/actuarial/basic.clj b/src/actuarial/basic.clj
new file mode 100644
index 0000000..4bb4573
--- /dev/null
+++ b/src/actuarial/basic.clj
@@ -0,0 +1,24 @@
+(ns actuarial.basic)
+(import [weka.core Statistics])
+
+(defn normalcdf "Normal CDF" [a] (Statistics/normalProbability a))
+(defn normalvar "Normal VAR" [a] (Statistics/normalInverse a))
+(def infinity Double/POSITIVE_INFINITY)
+
+(defn binomial-coefficient [n k]
+  (let [rprod (fn [a b] (reduce * (range a (inc b))))]
+    (/ (rprod (- n k -1) n) (rprod 1 k))))
+
+;; from https://gist.github.com/the-max0r/d5affa7c3c8a481733e5
+(defn rootNP
+  "Newton Raphson univariate root"
+  [f x]
+  (def step 1E-4)
+  (def eps 1E-8)
+  (let [m (/ (- (f (+ x step)) (f x)) step)
+        b (- (f x) (* m x))
+        x-0 (- (/ b m))]
+    (if (<= (Math/abs (- x x-0)) eps)
+      x-0
+      (recur f x-0))
+    ))
\ No newline at end of file
diff --git a/src/actuarial/core.clj b/src/actuarial/core.clj
new file mode 100644
index 0000000..98682eb
--- /dev/null
+++ b/src/actuarial/core.clj
@@ -0,0 +1,5 @@
+(ns actuarial.core (:require [actuarial.distributions]
+                             [actuarial.mixed-distributions]
+                             [actuarial.basic]
+                             [actuarial.measures]
+                             [actuarial.http]))
diff --git a/src/actuarial/distributions.clj b/src/actuarial/distributions.clj
new file mode 100644
index 0000000..1cb3f7f
--- /dev/null
+++ b/src/actuarial/distributions.clj
@@ -0,0 +1,1270 @@
+(ns actuarial.distributions (:use actuarial.basic))
+(import [weka.core Statistics])
+(use '(incanter core optimize))
+
+;; Methods
+
+(defmulti pdf "Probability density function" :Distribution)
+(defmulti pmf "Probability mass function" :Distribution)
+(defmulti survival "Survival function" :Distribution)
+(defmulti cdf "Cumulative distribution function" :Distribution)
+(defmulti rawmoments "Raw moments function" :Distribution)
+(defmulti mean "mean function" :Distribution)
+(defmulti variance "Variance function" :Distribution)
+(defmulti standard-deviation "Standard deviation function" :Distribution)
+(defmulti mgf "Moment generating function" :Distribution)
+(defmulti pgf "Probability generating function" :Distribution)
+(defmulti truncated-mean "Truncated mean function" :Distribution)
+(defmulti value-at-risk "Value at risk function" :Distribution)
+(defmulti tail-value-at-risk "Tail value at risk function" :Distribution)
+(defmulti stop-loss "Stop-loss function" :Distribution)
+(defmulti mean-excess "mean excess function" :Distribution)
+(defmulti limited-mean "Limited mean function" :Distribution)
+(defmulti standardize "Standardize a value to zero mean and unit variance" :Distribution)
+(defmulti describe "Describe a distribution" :Distribution)
+
+;; ## Default methods
+
+(defmethod survival :default [f x]
+  (- 1 (cdf f x)))
+
+(defmethod stop-loss :default [p d]
+  (- (truncated-mean p d)
+     (* d (survival p d))))
+
+(defmethod limited-mean :default [f d]
+  (- (mean f) (stop-loss f d)))
+
+(defmethod mean-excess :default [f d]
+  (/ (stop-loss f d) (survival f d)))
+
+(defmethod mean :default [f] (rawmoments f 1))
+
+(defmethod variance :default [f]
+  (- (rawmoments f 2) (pow (rawmoments f 1) 2)))
+
+(defmethod standard-deviation :default [f]
+  (sqrt (variance f)))
+
+(defmethod value-at-risk :default [f kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (case kappa
+    0 (get-in f [:Domain :min])
+    1 (get-in f [:Domain :max])
+    (rootNP #(- (cdf f %) kappa) (mean f))))
+
+(defmethod tail-value-at-risk :default [f kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [d (value-at-risk f kappa)]
+    (case kappa
+      0 (get-in f [:Domain :min])
+      1 (get-in f [:Domain :max])
+      (+ d (/ (stop-loss f d) (- 1 kappa))))))
+
+(defmethod standardize :default [f x]
+  (/ (- x (mean f)) (standard-deviation f)))
+
+(defmethod mgf :default [f t]
+  (+ 1 (reduce + (map #(/ (* (pow t %)
+                             (rawmoments f %))
+                          (factorial %))
+                      (range 1 100)))))
+
+;; Describe a distribution
+
+(defn in-domain [f x]
+  (let [min (get-in f [:Domain :min])
+        max (get-in f [:Domain :max])]
+    (<= min x max)
+  ))
+
+(defn describe [f x d t k]
+  {:distribution f
+   :x x
+   :deductible d
+   :degree t
+   :kappa k
+   :pdf  (try (pdf f x) (catch Exception e (println "Error in pdf!" e) 0))
+   :cdf  (try (cdf f x) (catch Exception e (println "Error in cdf!" e) 0))
+   :survival  (try (survival f x) (catch Exception e (println "Error in survival!" e) 0))
+   :mean (try (mean f) (catch Exception e (println "Error in mean!" e) 0))
+   :variance (try (variance f) (catch Exception e (println "Error in variance!" e) 0))
+   :truncated-mean (try (truncated-mean f d) (catch Exception e (println "Error in truncated-mean!" e) 0))
+   ;:value-at-risk (try (map #(value-at-risk f %) k) (catch Exception e (println "Error in value-at-risk !" e) 0))
+   ;:tail-value-at-risk (try (map #(tail-value-at-risk f %) k) (catch Exception e (println "Error in tail-value-at-risk!" e) 0))
+   :stop-loss (try (stop-loss f d) (catch Exception e (println "Error in stop-loss!" e) 0))
+   :mean-excess (try (mean-excess f d) (catch Exception e (println "Error in mean-excess!" e) 0))
+   :limited-mean (try (limited-mean f d) (catch Exception e (println "Error in limited-mean!" e) 0))
+   })
+
+;; ## Discrete distributions
+
+(defn arithmetic-dist
+  [h & prob]
+  {:pre [(> h 0)(= 1.0 (reduce + (vec prob)))]}
+  (let [n (dec (count (vec prob)))]
+  {:Distribution ::Arithmetic 
+   :Parameters {:h h :prob prob :n n}
+   :Domain {:min 0 :max n}
+   :Type ::Discrete}))
+
+(defn poisson-dist
+  [lambda]
+  {:pre (> lambda 0)}
+  {:Distribution ::Poisson 
+   :Parameters {:lambda lambda}
+   :Domain {:min 0 :max infinity }
+   :Type ::Discrete})
+
+(defn bernoulli-dist
+  [q]
+  {:pre [(<= 0 q 1)]}
+  {:Distribution ::Bernoulli
+   :Parameters {:q q}
+   :Domain {:min 0 :max 1}
+   :Type ::Discrete})
+
+(defn binomial-dist
+  [n q]
+  {:pre [(integer? n) (<= 0 q 1)]}
+  (if (= n 1)
+    (bernoulli-dist q)
+    ({:Distribution ::Binomial
+      :Parameters {:n n :q q}
+      :Domain {:min 0 :max n}
+      :Type ::Discrete})))
+
+(defn geometric-dist
+  [q]
+  {:pre [(< 0 q 1)]}
+   {:Distribution ::Geometric
+   :Parameters {:q q}
+   :Domain {:min 0 :max infinity}
+   :Type ::Discrete})
+
+(defn negative-binomial-dist
+  [r q]
+  {:pre [(<= 0 q 1)]}
+  (if (= r 1)
+    (geometric-dist q)
+  ({:Distribution ::NegativeBinomial
+    :Parameters {:r r :q q}
+    :Domain {:min 0 :max infinity}
+    :Type ::Discrete})))
+
+(defn negative-binomial-type2 
+  [r betap]
+  {:pre [(> r 0) (> betap 0)]}
+  (negative-binomial-dist 
+    r (/ 1 (inc betap))))
+
+(defn logarithmic-dist
+  [nu]
+  {:pre [(< 0 nu 1)]}
+  {:Distribution ::Logarithmic
+   :Parameters {:nu nu}
+   :Domain {:min 1 :max infinity}
+   :Type ::Discrete})
+
+;; ## Continuous distributions
+(defn uniform-dist 
+  ([a b] {:pre [(< a b)]}
+         {:Distribution ::Uniform 
+          :Parameters {:a a :b b}
+          :Domain {:min :max}
+          :Type ::Continuous})
+  ([b] (uniform-dist 0 b))
+  ([] (uniform-dist 0 1)))
+
+(defn exponential-dist 
+  [betap] 
+  {:pre [(> betap 0)]}
+  {:Distribution ::Exponential 
+   :Parameters{:beta betap}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn beta-dist 
+  [alpha betap] 
+  {:pre [(> alpha 0) (> betap 0)]}
+  {:Distribution ::Beta 
+   :Parameters {:alpha alpha :beta betap}
+   :Domain {:min 0 :max 1}
+   :Type ::Continuous})
+
+(defn gamma-dist
+  [alpha betap]
+  {:pre [(> alpha 0) (> betap 0)]}
+  {:Distribution ::Gamma 
+   :Parameters {:alpha alpha :beta betap :erlang false}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn erlang-dist
+  [n betap]
+  {:pre [(> n 0) (> betap 0) (integer? n)]}
+  (assoc-in (gamma-dist n betap) [:Parameters :erlang] true))
+
+(defn weibull-dist 
+  [tau betap] 
+  {:pre [(> tau 0) (> betap 0)]} 
+  {:Distribution ::Weibull 
+   :Parameters {:tau tau :beta betap}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn generalized-erlang-dist
+  [& betas]
+  {:pre [(every? #(> % 0) betas) (apply distinct? betas)]}
+  {:Distribution ::generalized-erlang 
+   :Parameters {:betas betas :n (count (vec betas))}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn gaussian-dist
+  ([mu sigma]
+    {:pre [(> sigma 0)]}
+    {:Distribution ::Gaussian 
+     :Parameters {:mu mu :sigma sigma}
+     :Domain {:min (- infinity) :max infinity}
+     :Type ::Continuous})
+  ([] (gaussian-dist 0 1)))
+
+(defn lognormal-dist
+  ([mu sigma]
+    {:pre [(> sigma 0)]}
+    {:Distribution ::Lognormal :Parameters {:mu mu :sigma sigma}
+     :Domain {:min 0 :max infinity}
+     :Type ::Continuous})
+  ([] (lognormal-dist 0 1)))
+
+(defn inverse-gaussian-dist
+  [mu betap]
+  {:pre [(>= betap 0)]}
+  {:Distribution ::InverseGaussian 
+   :Parameters {:mu mu :beta betap}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn pareto-dist
+  [alpha lambda]
+  {:pre [(> alpha 0) (> lambda 0)]}
+  {:Distribution ::Pareto 
+   :Parameters {:alpha alpha :lambda lambda}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn generalized-pareto-dist
+  [alpha lambda tau]
+  {:pre [(> alpha 0) (> lambda 0) (> tau 0)]}
+  {:Distribution ::GeneralizedPareto 
+   :Parameters {:alpha alpha :lambda lambda :tau tau}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn burr-dist
+  [alpha lambda tau]
+  {:pre [(> alpha 0) (> lambda 0) (> tau 0)]}
+  {:Distribution ::Burr
+   :Parameters {:alpha alpha :lambda lambda :tau tau}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn log-logistic-dist
+  [lambda tau]
+  {:pre (> tau 0)}
+  {:Distribution ::LogLogistic
+   :Parameters {:lambda lambda :tau tau}
+   :Domain {:min 0 :max infinity}
+   :Type ::Continuous})
+
+(defn student-dist
+  [nu]
+  {:pre (> nu 0)}
+  {:Distribution ::Student
+   :Parameters {:nu nu}
+   :Domain {:min (- infinity) :max infinity}
+   :Type ::Continuous})
+
+;; map of all distributions 
+(def distributions-map 
+  {"arithmetic" arithmetic-dist
+   "poisson" poisson-dist
+   "uniform" uniform-dist 
+   "exponential" exponential-dist 
+   "beta" beta-dist 
+   "gamma" gamma-dist 
+   "erlang" erlang-dist
+   "weibull" weibull-dist 
+   "generalized-erlang" generalized-erlang-dist
+   "gaussian" gaussian-dist
+   "lognormal" lognormal-dist
+   "inverse-gaussian" inverse-gaussian-dist
+   "pareto" pareto-dist 
+   "generalized-pareto" generalized-pareto-dist
+   "burr" burr-dist
+   "log-logistic" log-logistic-dist
+   "student" student-dist
+   })
+
+;; # Discrete distributions
+
+;; ## Arithmetic distribution
+
+(defmethod pmf ::Arithmetic [f x]
+  {:pre (in-domain f x)}
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])]
+    (nth prob (/ x h))))
+
+(defmethod cdf ::Arithmetic [f x]
+  {:pre (in-domain f x)}
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])]
+    (reduce + (take (inc (/ x h)) 
+                    (map #(pmf f (* % h)) 
+                         (iterate inc 0))))))
+
+(defmethod mean ::Arithmetic [f]
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])
+        n (get-in f [:Parameters :n])]
+    (reduce + (map #(* % h (nth prob %))
+                   (range (inc n))))))
+
+(defmethod variance ::Arithmetic [f]
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])
+        n (get-in f [:Parameters :n])]
+    (reduce + (map #(* (pow (- (* % h) 
+                               (mean f)) 2) 
+                       (nth prob %)) 
+                   (range (inc n))))))
+
+(defmethod mgf ::Arithmetic [f t]
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])
+        n (get-in f [:Parameters :n])]
+    (reduce + (map #(* (exp (* t h %)) 
+                       (nth prob %)) 
+                   (range (inc n))))))
+
+(defmethod pgf ::Arithmetic [f t]
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])
+        n (get-in f [:Parameters :n])]
+    (reduce + (map #(* (pow t (* h %)) 
+                       (nth prob %)) 
+                   (range (inc n))))))
+
+(defmethod truncated-mean ::Arithmetic [f d]
+  (let [prob (get-in f [:Parameters :prob]) 
+        h (get-in f [:Parameters :h])]
+    (reduce + (map #(* % h (nth prob %))
+                   (range (inc d))))))
+
+(defmethod value-at-risk ::Arithmetic [f kappa]
+  (first (filter #(> (cdf f %) kappa) (iterate inc 0))))
+
+(defmethod tail-value-at-risk ::Arithmetic [f kappa]
+  (let [h (get-in f [:Parameters :h])
+        k0 (value-at-risk f kappa)]
+    (/ (+ (- (mean f)
+             (reduce + (map #(* % h (pmf f (* % h)))
+                            (range 0 (inc k0)))))
+          (* k0 h (- (cdf f (* k0 h)) kappa)))
+       (- 1 kappa))))
+
+;; ## Poisson distribution
+
+(defmethod pmf ::Poisson [p k]
+  {:pre (in-domain p k)}
+  (let [lambda (get-in p [:Parameters :lambda])]
+    (/ (* (pow lambda k)
+          (exp (- lambda)))
+       (factorial k))))
+
+(defmethod cdf ::Poisson [p k]
+  {:pre (in-domain p k)}
+  (reduce + (map #(pmf %) (range (inc k)))))
+
+(defmethod mean ::Poisson [p] 
+  (let [lambda (get-in p [:Parameters :lambda])] 
+    lambda))
+
+(defmethod variance ::Poisson [p]   
+  (let [lambda (get-in p [:Parameters :lambda])] 
+    lambda))
+
+(defmethod pgf ::Poisson [p t] 
+  (let [lambda (get-in p [:Parameters :lambda])] 
+    (exp (* lambda (- t 1)))))
+
+(defmethod mgf ::Poisson [p t] 
+  (let [lambda (get-in p [:Parameters :lambda])] 
+    (pgf p (exp t))))
+
+
+;; ## Bernoulli distribution
+
+(defmethod pmf ::Bernoulli [b k]
+  (let [q (get-in b [:Parameters :q])]
+  (* (pow q k)
+     (pow (- 1 q) (- 1 k)))
+  ))
+
+(defmethod mean ::Bernoulli [b]
+  (let [q (get-in b [:Parameters :q])]
+    q))
+
+(defmethod variance ::Bernoulli [b]
+  (let [q (get-in b [:Parameters :q])]
+    (* q (- 1 q))))
+
+(defmethod mgf ::Bernoulli [b t]
+  (let [q (get-in b [:Parameters :q])]
+    (+ (* q (exp t)) (- 1 q))))
+
+(defmethod pgf ::Bernoulli [b t]
+  (let [q (get-in b [:Parameters :q])]
+    (+ (* q t) (- 1 q))))
+
+;; ## Binomial distribution
+
+(defmethod pmf ::Binomial [b k]
+  (let [n (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+  (* (binomial-coefficient n k)
+     (pow q k)
+     (pow (- 1 q) (- n k)))
+  ))
+
+(defmethod mean ::Binomial [b]
+  (let [n (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (* n q)))
+
+(defmethod variance ::Binomial [b]
+  (let [n (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (* n q (- 1 q))))
+
+(defmethod mgf ::Binomial [b t]
+  (let [n (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (pow (+ (* q (exp t)) (- 1 q)) n)))
+
+(defmethod pgf ::Binomial [b t]
+  (let [n (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (pow (+ (* q t) (- 1 q)) n)))
+
+;; ## Negative binomial distribution
+
+(defmethod pmf ::NegativeBinomial [b k]
+  (let [r (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (* (binomial-coefficient (+ r k 1) k )
+       (pow q r)
+       (pow (- 1 q) k))))
+
+(defmethod mean ::NegativeBinomial [b]
+  (let [r (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+  (/ (* r (- 1 q)) q)))
+
+(defmethod variance ::NegativeBinomial [b]
+  (let [r (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+  (/ (* r (- 1 q)) (pow q 2)))
+  )
+
+(defmethod mgf ::NegativeBinomial [b t]
+  (let [r (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (pow (/ q (- 1 (* (- 1 q) (exp t)))) r)
+  ))
+
+(defmethod pgf ::NegativeBinomial [b t]
+    (let [r (get-in b [:Parameters :n])
+        q (get-in b [:Parameters :q])]
+    (pow (/ q (- 1 (* (- 1 q) t))) r)
+  ))
+
+;; ## Geometric distribution
+
+(defmethod pmf ::Geometric [b k]
+  (let [q (get-in b [:Parameters :q])]
+    (* q (- 1 q))))
+
+(defmethod cdf ::Geometric [b k]
+  (let [q (get-in b [:Parameters :q])]
+    (- 1 (pow (- 1 q) (inc k)))))
+
+(defmethod mean ::Geometric [b]
+  (let [q (get-in b [:Parameters :q])]
+  (/ (- 1 q) q)))
+
+(defmethod variance ::Geometric [b]
+  (let [q (get-in b [:Parameters :q])]
+  (/ (- 1 q) (pow q 2))))
+
+(defmethod mgf ::Geometric [b t]
+  (let [q (get-in b [:Parameters :q])]
+    (/ q (- 1 (* (- 1 q) (exp t))))))
+
+(defmethod pgf ::Geometric [b t]
+    (let [q (get-in b [:Parameters :q])]
+    (/ q (- 1 (* (- 1 q) t)))))
+
+;; ## Logarithmic distribution
+
+(defmethod pmf ::Logarithmic [b k]
+  (let [nu (get-in b [:Parameters :nu])]
+    (/ (- (pow nu k)) (* (log (- 1 nu)) k)))) 
+
+(defmethod mean ::Logarithmic [b]
+  (let [nu (get-in b [:Parameters :nu])]
+  (/ (- nu) (* (log (- 1 nu)) (- 1 nu)))))
+
+(defmethod variance ::Logarithmic [b]
+  (let [nu (get-in b [:Parameters :nu])]
+    (/ (+ nu (log (- 1 nu))) 
+       (* (pow (- 1 nu) 2)
+          (pow (log (- 1 nu)) 2)))))
+
+(defmethod mgf ::Logarithmic [b t]
+  (let [nu (get-in b [:Parameters :nu])]
+    (/ (log (- 1 (* nu (exp t))))
+       (log (- 1 nu)))))
+
+(defmethod pgf ::Logarithmic [b t]
+    (let [nu (get-in b [:Parameters :nu])]
+    (/ (log (- 1 (* nu t)))
+       (log (- 1 nu)))))
+
+;; # Continuous distributions
+
+;; ## Uniform distribution
+
+(defmethod pdf ::Uniform [u x]
+  {:pre (in-domain u x)}
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (/ 1 (- b a))))
+
+(defmethod cdf ::Uniform [u x] 
+  {:pre (in-domain u x)}
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (min (max 0 (/ (- x a) (- b a))) 1)))
+
+(defmethod rawmoments ::Uniform [u k] 
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (/ (- (pow b (inc k)) (pow a (inc k)))
+       (* (inc k) (- b a)))))
+
+(defmethod mgf ::Uniform [u t] 
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (/ (- (Math/exp (* b t)) 
+          (Math/exp (* a t)))
+       (* (- b a) t))))
+
+(defmethod truncated-mean ::Uniform [u d] 
+  {:pre (< d (get-in u [:Parameters :b]))}
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (/ (- (pow d 2) (pow a 2)) 
+       (* 2 (- b a)))))
+
+(defmethod value-at-risk ::Uniform [u kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [a (get-in u [:Parameters :a]) 
+        b (get-in u [:Parameters :b])]
+    (+ a (* (- b a) kappa))))
+
+;; ## Exponential distribution
+
+(defmethod pdf ::Exponential [e x]
+  {:pre (in-domain e x)}
+  (let [betap (get-in e [:Parameters :beta])]
+    (* betap (exp (- (* betap x))))))
+
+(defmethod cdf ::Exponential [e x]
+  {:pre (in-domain e x)}
+  (let [betap (get-in e [:Parameters :beta])]
+    (- 1 (exp (- (* betap x))))))
+
+(defmethod mean ::Exponential [e]
+  (let [betap (get-in e [:Parameters :beta])]
+    (/ 1 betap)))
+
+(defmethod variance ::Exponential [e]
+  (let [betap (get-in e [:Parameters :beta])]
+    (/ 1 (pow betap 2))))
+
+(defmethod mgf  ::Exponential [e t]
+  {:pre [(< t (get-in e [:Parameters :beta]))]}
+  (let [betap (get-in e [:Parameters :beta])]
+    (/ betap (- betap t))))
+
+(defmethod rawmoments ::Exponential [e k]
+  (let [betap (get-in e [:Parameters :beta])]
+    (* (pow (/ 1 betap) k) (factorial k))))
+
+(defmethod stop-loss ::Exponential [e d]
+  (* (mean e) (survival e d)))
+
+(defmethod truncated-mean ::Exponential [e d]
+  (- (* (limited-mean e d))
+     (* d (survival e d))))
+
+(defmethod value-at-risk  ::Exponential [e kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [betap (get-in e [:Parameters :beta])]
+    (- (* (/ 1 betap)
+          (log (- 1 kappa))))))
+
+;; ## Beta functions
+
+(defn beta-complete
+  "Complete beta function"
+  [alpha betap]
+  {:pre [(> alpha 0) (> betap 0)]}
+  (/ (* (gamma alpha) 
+        (gamma betap))
+     (gamma (+ alpha betap))))
+
+(defn beta-incomplete 
+  "Incomplete beta function"
+  [alpha betap x]
+  {:pre [(> alpha 0) (> betap 0)]}
+  (* (Statistics/incompleteBeta alpha betap x) 
+     (beta-complete alpha betap)))
+
+;; ## Beta distribution
+
+(defmethod pdf ::Beta [b x]
+  {:pre (in-domain b x)}
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (* (/ 1 (beta-complete alpha betap)) 
+       (pow x (- alpha 1)) (pow (- 1 x)(- betap 1)))))
+
+(defmethod cdf ::Beta [b x]
+  {:pre (in-domain b x)}
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (defn term-of-sum [alpha betap j]
+      (* (/ (factorial (+ alpha betap (- 1))) 
+            (* (factorial j)
+               (factorial (- (+ alpha betap) j 1))))
+         (pow x j)
+         (pow (- 1 x)
+              (-(+ alpha betap) 1 j))))
+    (if (and (integer? alpha) (integer? betap))
+      (reduce + (map #(term-of-sum alpha betap %) 
+                     (range alpha (+ alpha betap))))
+      (/ (beta-incomplete alpha betap x) 
+         (beta-complete alpha betap)))))
+
+(defn beta-mgf-term-of-product
+  [alpha betap j]
+  {:pre [(> alpha 0) (> betap 0)]}
+  (/ (+ alpha j)
+     (+ alpha betap j)))
+
+(defn beta-mgf-innerproduct
+  [alpha betap k]
+  {:pre [(> alpha 0) (> betap 0)]}
+  (reduce * (map #(beta-mgf-term-of-product alpha betap %) 
+                 (range k))))
+
+(defn beta-mgf-term-of-sum
+  [alpha betap t k]
+  {:pre [(> alpha 0) (> betap 0)]}
+  (* (beta-mgf-innerproduct alpha betap k) 
+     (/ (pow t k) 
+        (factorial k))))
+
+(defn beta-mgf-sums [alpha betap t] 
+  {:pre [(> alpha 0) (> betap 0)]}
+  (map #(beta-mgf-term-of-sum alpha betap t %) 
+       (rest (range))))
+
+(defmethod mgf ::Beta [b t]
+  {:pre []}
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (reduce + (take 100 (beta-mgf-sums alpha betap t)))))
+
+(defmethod rawmoments ::Beta [b k]
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (/(*(gamma (+ alpha k))
+        (gamma (+ alpha betap)))
+      (*(gamma alpha)
+        (gamma (+ alpha betap k))))))
+
+(defmethod truncated-mean ::Beta [b d]
+  {:pre [(>= d 0) (<= d 1)]}
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (if (= betap 1)
+      (/ (* alpha (pow d (+ alpha 1)))
+         (+ alpha 1))
+      (if (= alpha 1)
+        (+ (- 0 (* d (pow (- 1 d) betap))) 
+           (/ (- 1 (pow (- 1 d) (+ betap 1)))
+              (+ betap 1)))
+        (* (/ alpha 
+              (+ alpha betap))
+           (cdf (beta-dist (+ alpha 1) betap) d))))))
+
+(defmethod value-at-risk ::Beta [b kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [alpha (get-in b [:Parameters :alpha]) 
+        betap (get-in b [:Parameters :beta])]
+    (if (= betap 1)
+      (pow kappa (/ 1 alpha))
+      (if (= alpha 1)
+        (- 1 (pow (- 1 kappa) (/ 1 betap)))
+        (rootNP #(- (cdf b %) kappa) (mean b))))))
+
+;; ## Gamma and Erlang distributions
+
+(defmethod pdf ::Gamma [g x]
+  {:pre (in-domain g x)}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (* (/ (pow betap alpha)
+          (gamma alpha))
+       (pow x (- alpha 1))
+       (exp (- (* betap x))))))
+
+(defn incompletegamma [s x]
+  (* (pow x s)
+     (gamma s)
+     (exp (- x))
+     (reduce + (map #(/ (pow x %) (gamma (+ s 1 %))) (range 100)))))
+
+(defn upperincompletegamma  [s x]
+  (- (gamma s)
+     (incompletegamma s x)))
+
+(defmethod cdf ::Gamma [g x]
+  {:pre (in-domain g x)}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (if (integer?  alpha)
+      (- 1 (* (exp (- (* betap x)))
+              (reduce + (map #(/ (pow (* betap x) %)
+                                 (factorial %)) 
+                             (range  alpha)))))
+      (/ (incompletegamma  alpha (* betap x))
+         (gamma  alpha)))))
+
+(defmethod survival ::Gamma [g x]
+  {:pre (in-domain g x)}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (if (integer?  alpha)
+      (- 1 (cdf g x))
+      (/ (upperincompletegamma
+           alpha
+           (* betap x))
+         (gamma  alpha)))))
+
+
+(defmethod mgf ::Gamma [g t]
+  {:pre (< t (:beta g))}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (pow (/ betap(- betap t))  alpha)))
+
+(defmethod rawmoments ::Gamma [g k]
+  {:pre (integer? k)}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (/ (reduce * (map #(+  alpha %) (range k)))
+       (pow betap k))))
+
+(defmethod truncated-mean ::Gamma [g d]
+  {:pre (> d 0)}
+  (let [alpha (get-in g [:Parameters :alpha]) 
+        betap (get-in g [:Parameters :beta])]
+    (* (/ alpha betap)
+       (cdf (gamma-dist (+  alpha 1) betap) d))))
+
+;; ## Auxiliairy distribution for the Weibull method definitions
+(defn auxiliairy-gamma-weibull [w]
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (gamma-dist 
+      (+ 1 (/ 1 tau)) 
+      (pow betap tau))))
+
+;; ## Weibull distribution
+
+(defmethod pdf ::Weibull [w x]
+  {:pre (in-domain w x)}
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (* betap 
+       tau 
+       (pow (* betap x)
+            (- tau 1))
+       (exp (- (pow (* betap x)
+                    tau))))))
+
+(defmethod cdf ::Weibull [w x]
+  {:pre (in-domain w x)}
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (- 1 (exp (- (pow (* betap x) tau))))))
+
+(defmethod rawmoments ::Weibull [w k]
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (/ (gamma (+ 1 (/ k tau))) 
+       (pow betap k))))
+
+(defmethod mgf ::Weibull [w k] 
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (reduce + (map #(* (rawmoments w %) 
+                       (/ (pow tau %) 
+                          (* (pow betap %) 
+                             (factorial %)))) 
+                   (range 100)))))
+
+(defmethod truncated-mean ::Weibull [w d]
+  (let [tau (get-in w [:Parameters :tau])]
+    (* (mean w) 
+       (cdf (auxiliairy-gamma-weibull w)
+            (pow d tau)))))
+
+(defmethod value-at-risk ::Weibull [w kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [betap (get-in w [:Parameters :beta]) 
+        tau (get-in w [:Parameters :tau])]
+    (/ (pow (- (log (- 1 kappa))) 
+            (/ 1 tau)) betap)))
+
+;; ## Generalized Erlang Distribution
+;; [reference: R package sdprisk]
+;; (htaup://www.inside-r.org/packages/cran/sdprisk/docs/hypoexp)
+;; *Helper functions*
+
+;; Inner big-product
+(defn helper-product-generalized-erlang [i betas]
+  (let [nbeta (nth betas i)]
+    (reduce * (map #(/ % (- % nbeta))
+                   (filter #(not (= % nbeta)) betas)))))
+;; Big-sum for functions of generalized erlang distribution
+(defn helper-big-sum-generalized-erlang [i x betas f]
+  (* (helper-product-generalized-erlang i betas)
+     (f i x betas)))
+
+;; *methods*
+
+(defmethod pdf ::generalized-erlang [g x]
+  {:pre (in-domain g x)}
+  (let [betas (get-in g [:Parameters :betas])
+        n (get-in g [:Parameters :n])]
+    (reduce + (map #(helper-big-sum-generalized-erlang 
+                      % x (vec betas)
+                      (fn [i x betas] 
+                        (let [nbeta (nth betas i)]
+                          (* nbeta 
+                             (exp (- (* x nbeta)))))))
+                   (range n)))))
+
+(defmethod cdf ::generalized-erlang [g x]
+  {:pre (in-domain g x)}
+  (let [betas (get-in g [:Parameters :betas])
+        n (get-in g [:Parameters :n])]
+    (reduce + (map #(helper-big-sum-generalized-erlang 
+                      % x (vec betas)
+                      (fn [i x betas] 
+                        (- 1 (exp (- (* x (nth betas i)))))))
+                   (range n)))))
+
+(defmethod mgf ::generalized-erlang [g t]
+  (let [betas (get-in g [:Parameters :betas])]
+    (reduce * (map #(/ % (- % t)) betas))))
+
+(defmethod rawmoments ::generalized-erlang [g k]
+  (let [betas (get-in g [:Parameters :betas])]
+    (reduce * (map #(/ (gamma (+ k 1)) (pow % k)) betas))))
+
+(defmethod truncated-mean ::generalized-erlang [g d]
+  (let [betas (get-in g [:Parameters :betas])
+        n (get-in g [:Parameters :n])]
+    (reduce + (map #(helper-big-sum-generalized-erlang 
+                      % d (vec betas) 
+                      (fn [i d betas]
+                        (let [nbeta (nth betas i)]
+                          (+ (* (- d) (exp (-(* nbeta d))))
+                             (/ (- 1 (exp (-(* nbeta d))))
+                                nbeta)))))
+                   (range n)))))
+
+;; ## Gaussian Distribution
+
+(defmethod pdf ::Gaussian [g x]
+  {:pre (in-domain g x)}
+  (let [sigma (get-in g [:Parameters :sigma])]
+    (/ (exp (- (* 0.5 (pow (standardize g x) 2))))
+       (* sigma (sqrt (* 2 Math/PI))))))
+
+(defmethod cdf ::Gaussian [g x]
+  {:pre (in-domain g x)}
+  (normalcdf (standardize g x)))
+
+(defmethod mean ::Gaussian [g]
+  (let [mu (get-in g [:Parameters :mu])]
+    mu))
+
+(defmethod variance ::Gaussian [g] 
+  (let [sigma (get-in g [:Parameters :sigma])]
+    (pow sigma 2)))
+
+(defmethod mgf ::Gaussian [g t] 
+  (let [mu (get-in g [:Parameters :mu])
+        sigma (get-in g [:Parameters :sigma])]
+    (exp (+ (* t mu)
+            (* 0.5 (pow t 2) (pow sigma 2))))))
+
+(defn sigmasq-times-pdf-helper [g d]
+  (let [sigma (get-in g [:Parameters :sigma])]
+    (* (pow sigma 2) (pdf g d))))
+
+(defmethod truncated-mean ::Gaussian [g d]
+  (let [mu (get-in g [:Parameters :mu])]
+    (- (* mu (cdf g d))
+       (sigmasq-times-pdf-helper g d))))
+
+(defmethod value-at-risk ::Gaussian [g kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [mu (get-in g [:Parameters :mu])
+        sigma (get-in g [:Parameters :sigma])]
+    (+ mu 
+       (* sigma
+          (normalvar kappa)))))
+
+;; ## Lognormal distribution
+
+(defmethod standardize ::Lognormal [l x]
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (/ (- (log x) mu (pow sigma 2)) sigma)))
+
+(defmethod pdf ::Lognormal [l x]
+  {:pre (in-domain l x)}
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (/ (pdf (gaussian-dist mu sigma) (log x)) x)))
+
+(defmethod cdf ::Lognormal [l x]
+  {:pre (in-domain l x)}
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (cdf (gaussian-dist mu sigma) (log x))))
+
+(defmethod rawmoments ::Lognormal [l k]
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (exp (+ (* k mu) 
+            (/ (* (pow k 2) 
+                  (pow sigma 2)) 2)))))
+
+(defmethod truncated-mean ::Lognormal [l d]
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (* (exp (+ mu 
+               (/ (pow sigma 2) 2)))
+       (normalcdf (standardize l d)))))
+
+(defmethod value-at-risk ::Lognormal [l kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [mu (get-in l [:Parameters :mu])
+        sigma (get-in l [:Parameters :sigma])]
+    (exp (+ mu 
+            (* sigma 
+               (value-at-risk 
+                 (gaussian-dist) 
+                 kappa))))))
+
+;; ## Inverse gaussian distribution
+
+(defn e2mb [i]     
+  (let [mu (get-in i [:Parameters :mu])
+        betap (get-in i [:Parameters :beta])]
+    (exp (/ (* 2 mu) betap))))
+
+(defn helper1 [i x s]
+  (let [mu (get-in i [:Parameters :mu])
+        betap (get-in i [:Parameters :beta])]
+    (normalcdf (* (sqrt (/ 1 (* betap x))) (- (* x s) mu)))))
+
+(defmethod pdf ::InverseGaussian [i x]
+  {:pre (in-domain i x)}
+  (let [mu (get-in i [:Parameters :mu])
+        betap (get-in i [:Parameters :beta])]
+    (/ (* mu (exp (- (/ (pow (- x mu) 2)
+                        (* 2 betap x)))))
+       (sqrt (* 2 Math/PI betap (pow x 3))))))
+
+(defmethod cdf ::InverseGaussian [i x]
+  {:pre (in-domain i x)}
+  (+ (helper1 i x 1)
+     (* (e2mb i)
+        (helper1 i x -1))))
+
+(defmethod mean ::InverseGaussian [i]
+  (let [mu (get-in i [:Parameters :mu])]
+    mu))
+
+(defmethod variance ::InverseGaussian [i]
+  (let [mu (get-in i [:Parameters :mu])
+        betap (get-in i [:Parameters :beta])]
+    (* mu betap)))
+
+(defmethod mgf ::InverseGaussian [i t]
+  (let [mu (get-in i [:Parameters :mu])
+        betap (get-in i [:Parameters :beta])]
+    (exp (* (/ mu beta) 
+            (- 1 (sqrt (- 1 (* 2 beta t))))))))
+
+(defmethod truncated-mean ::InverseGaussian [i d]
+  (let [mu (get-in i [:Parameters :mu])]
+    (- d
+       (* (- (* 2 d) mu) (helper1 i d 1))
+       (* (+ (* 2 d) mu) 
+          (e2mb i)
+          (helper1 i d -1)))))
+
+;; ## Pareto distribution
+
+(defmethod pdf ::Pareto [p x]
+  {:pre (in-domain p x)}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])]
+    (/ (* alpha (pow lambda alpha)) 
+       (pow (+ lambda x) (inc alpha)))))
+
+(defmethod cdf ::Pareto [p x]
+  {:pre (in-domain p x)}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])]
+    (- 1 (pow (/ lambda (+ lambda x)) alpha))))
+
+(defmethod rawmoments ::Pareto [p k]
+  {:pre (< k (get-in p [:Parameters :alpha]))}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])]
+    (if (and (< k alpha) (integer? k))
+      (/ (* (pow lambda k) (factorial k))
+         (reduce * (map #(- alpha %) (range 1 (inc k)))))
+      (if (and (< -1 k) (< k alpha))
+        (/ (* (pow lambda k) 
+              (gamma (inc k))
+              (gamma (- alpha k)))
+           (gamma alpha))
+        0))))
+
+(defmethod truncated-mean ::Pareto [p d]
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])]
+    (- (* (mean p) (cdf (pareto-dist (dec alpha) lambda) d))
+       (* d (survival p d)))))
+
+(defmethod value-at-risk ::Pareto [p kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])]
+    (* lambda (dec (pow (- 1 kappa) (- (/ 1 alpha)))))))
+
+;; ## Generalized Pareto 
+
+(defmethod pdf ::GeneralizedPareto [p x]
+  {:pre (in-domain p x)}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])
+        tau (get-in p [:Parameters :tau])]
+    (/ (* (gamma (+ alpha tau))
+          (pow lambda alpha)
+          (pow x (dec tau)))
+       (* (gamma alpha)
+          (gamma tau)
+          (pow (+ lambda x) (+ alpha tau))))))
+
+(defmethod cdf ::GeneralizedPareto [p x]
+  {:pre (in-domain p x)}
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])
+        tau (get-in p [:Parameters :tau])]
+    (cdf (beta-dist tau alpha) (/ x (+ lambda x)))))
+
+(defmethod rawmoments ::GeneralizedPareto [p k]
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])
+        tau (get-in p [:Parameters :tau])]
+    (* (pow lambda k) 
+       (/ (reduce * (map #(+ tau %) (range k)))
+          (reduce * (map #(- alpha %) (range 1 (inc k))))))))
+
+(defmethod truncated-mean ::GeneralizedPareto [p d]
+  (let [alpha (get-in p [:Parameters :alpha])
+        lambda (get-in p [:Parameters :lambda])
+        tau (get-in p [:Parameters :tau])]
+    (* (mean p) 
+       (cdf (beta-dist (inc tau) (dec alpha)) 
+            (/ d (+ lambda d))))))
+
+;; ## Burr
+
+(defmethod pdf ::Burr [b x] 
+  {:pre (in-domain b x)}
+  (let [alpha (get-in b [:Parameters :alpha])
+        lambda (get-in b [:Parameters :lambda])
+        tau (get-in b [:Parameters :tau])]
+    (/ (* alpha tau (pow lambda alpha) (pow x (dec tau)))
+       (pow (+ lambda (pow x tau))
+            (inc alpha)))))
+
+(defmethod cdf ::Burr [b x] 
+  {:pre (in-domain b x)}
+  (let [alpha (get-in b [:Parameters :alpha])
+        lambda (get-in b [:Parameters :lambda])
+        tau (get-in b [:Parameters :tau])]
+    (- 1 (pow (/ lambda 
+                 (+ lambda 
+                    (pow x tau)))
+              alpha))))
+
+(defmethod rawmoments ::Burr [b k]
+  {:pre [(< (-(get-in b [:Parameters :tau])) k) 
+         (< k (* (get-in b [:Parameters :alpha])
+                 (get-in b [:Parameters :tau])))]}
+  (let [alpha (get-in b [:Parameters :alpha])
+        lambda (get-in b [:Parameters :lambda])
+        tau (get-in b [:Parameters :tau])]
+    (/ (* (pow lambda (/ k tau))
+          (gamma (inc (/ k tau)))
+          (gamma (- alpha (/ k tau))))
+       (gamma alpha))))
+
+(defmethod truncated-mean ::Burr [b d]
+  (let [alpha (get-in b [:Parameters :alpha])
+        lambda (get-in b [:Parameters :lambda])
+        tau (get-in b [:Parameters :tau])]
+    (* (mean b) 
+       (cdf (beta-dist (inc (/ 1 tau))(- alpha (/ 1 tau)))
+            (/ (pow d tau)(+ lambda (pow d tau)))))))
+
+(defmethod value-at-risk ::Burr [b kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [alpha (get-in b [:Parameters :alpha])
+        lambda (get-in b [:Parameters :lambda])
+        tau (get-in b [:Parameters :tau])]
+    (pow (* lambda 
+            (dec (pow (- 1 kappa)
+                      (- (/ 1 alpha)))))
+         (/ 1 tau))))
+
+;; ## Log-logistique
+(defmethod pdf ::LogLogistic [l x]
+  {:pre (in-domain l x)}
+    (let [lambda (get-in l [:Parameters :lambda])
+        tau (get-in l [:Parameters :tau])]
+    (/ (* tau
+          (pow x (dec tau)))
+       (pow (+ (pow lambda tau) 
+               (pow x tau))
+            2))))
+
+(defmethod cdf ::LogLogistic [l x]
+  {:pre (in-domain l x)}
+  (let [px (pow x (get-in l [:Parameters :tau])) 
+        pl (pow (get-in l [:Parameters :lambda])
+                (get-in l [:Parameters :tau]))]
+    (/ px (+ pl px))))
+
+(defmethod rawmoments ::LogLogistic [l k]
+  {:pre (< (abs k) (get-in l [:Parameters :tau]))}
+    (let [lambda (get-in l [:Parameters :lambda])
+        tau (get-in l [:Parameters :tau])]
+    (* (pow lambda k)
+       (gamma (inc (/ k tau)))
+       (gamma (- 1 (/ k tau))))))
+
+(defmethod truncated-mean ::LogLogistic [l d]
+  {:pre (> (get-in l [:Parameters :tau]) 1)}
+    (let [lambda (get-in l [:Parameters :lambda])
+        tau (get-in l [:Parameters :tau])]
+    (* (mean l) 
+       (cdf (beta-dist (inc (/ 1 tau))(- 1 (/ 1 tau)))
+            (/ (pow d tau)(+ lambda (pow d tau)))))))
+
+(defmethod value-at-risk ::LogLogistic [l kappa]
+  {:pre [(<= 0 kappa 1)]}
+  (let [lambda (get-in l [:Parameters :lambda])
+        tau (get-in l [:Parameters :tau])]
+    (* lambda (pow (dec (/ 1 kappa))
+                   (- (/ 1 tau))))))
+
+;; ## Student
+
+(defmethod pdf ::Student [s x]
+  {:pre (in-domain s x)}
+  (let [nu (get-in s [:Parameters :nu])]
+    (case nu
+      1 (/ 1 (* Math/PI (+ 1 (pow x 2))))
+      2 (/ 1 (pow (+ 2 (pow x 2)) (/ 3 2)))
+      (/ (* (gamma (/ (+ 1 nu) 2)) 
+            (pow (+ 1 (/ (pow x 2) nu)) (- (/ (+ nu 1) 2))))
+         (* (sqrt (* nu Math/PI)) 
+            (gamma (/ nu 2)))))))
+
+(defmethod cdf ::Student [s x]
+  {:pre (in-domain s x)}
+  (let [nu (get-in s [:Parameters :nu])
+        corepart (/ (cdf (beta-dist (/ nu 2) 0.5) 
+                  (/ nu (+ nu (pow x 2)))) 2)]
+    (case nu
+      1 (+ (/ 1 2) (/ (Math/atan x) Math/PI))
+      2 (/ (+ 1 (/ x (sqrt (+ 2 (pow x 2))))) 2)
+      (cond
+      (< x 0.0) corepart
+      (= x 0.0) 0.5
+      (> x 0.0) (- 1 corepart)
+      ))))
+
+(defmethod rawmoments ::Student [s k]
+  {:pre [(< 0 k (get-in s [:Parameters :nu]))]}
+  (let [nu (get-in s [:Parameters :nu])]
+    (cond 
+      (odd? k) 0
+      (even? k) (/ (* (gamma (/ (inc k) 2)) 
+                      (gamma (/ (- nu k) 2)) 
+                      (pow nu (/ k 2)))
+                   (* (sqrt Math/PI) (gamma (/ nu 2))))
+      )))
+
+(defmethod truncated-mean ::Student [s d]
+  (let [nu (get-in s [:Parameters :nu])
+        main-part (* (sqrt (/ nu Math/PI))
+                     (/ (gamma (/ (inc nu) 2)) (gamma (/ nu 2)))
+                     (pow (inc (/ (pow d 2) nu))
+                          (/ (- 1 nu) 2)))
+        ]
+  (cond
+    (< d 0) main-part
+    (> d 0) (- main-part)))
+  )
+
+
diff --git a/src/actuarial/http.clj b/src/actuarial/http.clj
new file mode 100644
index 0000000..78e6e8d
--- /dev/null
+++ b/src/actuarial/http.clj
@@ -0,0 +1,50 @@
+(ns actuarial.http 
+  (:use ring.adapter.jetty)
+  (:use ring.middleware.params)
+  (:use actuarial.distributions))
+
+(require 'ring.middleware.stacktrace)
+
+;(describe (build-distribution (resolve (symbol (str "gaussian" "-dist"))) [1 3]) 0.5 0.25 3)
+
+(defn wrap-spy [handler]
+  (fn [request]
+    (println "-------------------------------")
+    (println "Incoming Request:")
+    (clojure.pprint/pprint request)
+    (let [response (handler request)]
+      (println "Outgoing Response Map:")
+      (clojure.pprint/pprint response)
+      (println "-------------------------------")
+      response)))
+
+(defmacro get-map-dist [map name param]
+  `(apply (get ~map ~name ) ~param))
+
+(defn handler [req]
+  (let [distr-name (get-in req [:query-params "dist"])
+        distr-params (read-string (get-in req [:query-params "params"]))
+        distr-x (read-string (get-in req [:query-params "x"] "0"))
+        distr-d (read-string (get-in req [:query-params "d"] "0"))
+        distr-t (read-string (get-in req [:query-params "t"] "1"))
+        distr-map (get-map-dist distributions-map distr-name distr-params)
+        sortie (describe distr-map distr-x distr-d distr-t)
+        ]
+    (println distr-name distr-params)
+    {:status 200
+     :headers {"Content-Type" "text/html"}
+     :body (str "<p>" sortie "<p>")}
+  ))
+
+(def app
+  (-> #'handler
+    (ring.middleware.stacktrace/wrap-stacktrace)
+    (wrap-spy)
+    (wrap-params)
+    ))
+
+(defonce server (ring.adapter.jetty/run-jetty #'app {:port 8080 :join? false}))
+
+(.stop server)
+
+(.start server)
\ No newline at end of file
diff --git a/src/actuarial/measures.clj b/src/actuarial/measures.clj
new file mode 100644
index 0000000..60dd623
--- /dev/null
+++ b/src/actuarial/measures.clj
@@ -0,0 +1,6 @@
+(ns actuarial.measures (:use actuarial.basic))
+(use '(incanter core))
+
+(defn node-entropy [& values]
+  (- (reduce + (map #(* % (/ (log %) (log 2)))
+                    (map #(/ % (reduce + values)) values)))))
diff --git a/src/actuarial/mixed_distributions.clj b/src/actuarial/mixed_distributions.clj
new file mode 100644
index 0000000..2244e80
--- /dev/null
+++ b/src/actuarial/mixed_distributions.clj
@@ -0,0 +1,32 @@
+(ns actuarial.mixed-distributions (:use actuarial.distributions))
+
+(defn mixedDist
+  [prob-dist cont-dist]
+  {:Distribution ::Mixed
+   :SubDistributions {:prob-dist prob-dist
+                      :cont-dist cont-dist}
+   :type ::Mixed})
+
+(defmacro getmixed [f m & params]
+  `(let [probs# (get-in ~m [:SubDistributions :prob-dist :Parameters :prob])
+         dists# (get-in ~m [:SubDistributions :cont-dist])]
+     (reduce + (map #(* %1 (~f %2 ~@params)) probs# dists#))))
+
+(defmethod pdf ::Mixed [m x]
+  (getmixed pdf m x))
+
+(defmethod cdf ::Mixed [m x]
+  (getmixed cdf m x))
+
+(defmethod mgf ::Mixed [m t]
+  (getmixed mgf m t))
+
+(defmethod rawmoments ::Mixed [m k]
+  (getmixed rawmoments m k))
+
+(defmethod truncated-mean ::Mixed [m d]
+  (getmixed truncated-mean m d))
+
+
+
+
diff --git a/test/actuarial/test/core.clj b/test/actuarial/test/core.clj
new file mode 100644
index 0000000..cee153a
--- /dev/null
+++ b/test/actuarial/test/core.clj
@@ -0,0 +1,123 @@
+(ns actuarial.core-test
+  (:require [clojure.test :refer :all]
+            [actuarial.core :refer :all]
+            [actuarial.distributions :as dist]
+            [actuarial.mixed-distributions :as mix]))
+;; # Discrete distributions
+
+;; ## Arithmetic distribution
+
+(defmacro dbg [& body]
+  `(let [x# ~@body]
+     (println (str "dbg: " (quote ~@body) "=" x#))
+     x#))
+
+(defn close? [tolerance x y]
+  (< (- x y) tolerance))
+
+(defn =? [x y] (partial close? 0.000000001))
+
+(deftest arith-dist
+  (let [arith1 (dist/arithmetic-dist 1 0.2 0.3 0.4 0.1)]
+    (testing "Arithmetic Distribution"
+           (is (=? (dist/pmf arith1 3) 0.1))
+           (is (=? (dist/cdf arith1 2) 0.9))
+           (is (=? (dist/survival arith1 2) 0.1))
+           (is (=? (dist/mean arith1) 1.4))  
+           (is (=? (dist/variance arith1) 1.4))
+           (is (=? (dist/value-at-risk arith1 0.6) 2))
+           (is (=? (dist/tail-value-at-risk arith1 0.6) 2.25))
+           )))
+
+;; ## Poisson distribution
+
+(deftest poisson-dist
+  (let [poisson1 (dist/poisson-dist 3)
+        =? (partial close? 0.00001)]
+  (testing "Poisson distribution"
+           (is (=? (dist/pmf poisson1 0) (Math/exp -3)))
+           (is (=? (dist/mean poisson1) 3))
+           (is (=? (dist/variance poisson1) 3))+
+           (is (=? (dist/pgf poisson1 2) (Math/exp 3)))
+           (is (=? (dist/mgf poisson1 2) (Math/exp(* 3 (- (Math/exp 2) 1)))))
+           )))
+
+;; ## Mixed Distributions
+
+;; tests
+
+(def dist1 (dist/arithmetic-dist 1 0.4 0.6))
+(def dist2 (map #(apply dist/exponential-dist %) '([0.4] [0.5])))
+(def mix1 (mix/mixedDist dist1 dist2))
+(println 
+  (dist/pdf mix1 0.25)
+  (dist/cdf mix1 0.25)
+  (dist/mgf mix1 0.1)
+  (dist/rawmoments mix1 3))
+
+(dist/describe mix1 0.25 0.5 2 0.99)
+
+;; # Continuous distributions
+
+;; Uniform distribution
+(def uniform1 (dist/uniform-dist 1.0 3.0))
+(println (dist/describe uniform1 2.0 2.5 2.0 [0.95]))
+
+;; Exponential distribution
+(def exp1 (dist/exponential-dist 2.0))
+(println (dist/describe exp1 2.0 2.5 1.0 [0.95]))
+
+;; Beta distribution
+(def beta1 (dist/beta-dist 1 3))
+(println (dist/describe beta1 0.75 0.25 2 [0.95]))
+
+;; Gamma distribution
+(def gamma1 (dist/gamma-dist 2.001 4.999))
+(println (dist/describe gamma1 0.75 0.25 2 [0.95]))
+
+;; Erlang distribution
+(def erlang1 (dist/erlang-dist 2 5))
+(println (dist/describe erlang1 0.75 0.25 2 [0.95]))
+
+;; Weibull distribution
+(def weibull1 (dist/weibull-dist 2.2 3.1))
+(println (dist/describe weibull1 0.75 0.25 2 [0.95]))
+
+;; Generalized Erlang distribution
+(def generlang1 (dist/generalized-erlang-dist 2.1 5 0.25))
+(println (dist/describe generlang1 0.75 0.25 2 [0.95]))
+
+;; Gaussian distribution
+(def gaussian1 (dist/gaussian-dist 1.5 0.5))
+(println (dist/describe gaussian1 0.75 0.25 2 [0.95]))
+
+;;Lognormal distribution
+(def lognormal1 (dist/lognormal-dist 1.5 0.5))
+(println (dist/describe lognormal1 0.75 0.25 2 [0.95]))
+
+;;Inverse Gaussian distribution
+(def inversegaussian1 (dist/inverse-gaussian-dist 1.5 0.5))
+(println (dist/describe inversegaussian1 0.75 0.25 2 [0.95]))
+
+;;Pareto distribution
+(def pareto1 (dist/pareto-dist 4 3))
+(println (dist/describe pareto1 0.75 0.25 2 [0.95]))
+
+;;Generalized Pareto distribution
+(def generalizedpareto1 (dist/generalized-pareto-dist 4 3 1))
+(println (dist/describe generalizedpareto1 0.75 0.25 2 [0.95]))
+
+;;Burr distribution
+(def burr1 (dist/burr-dist 4 3 1))
+(println (dist/describe burr1 0.75 0.25 2 [0.95]))
+
+;;Log-Logistic distribution
+(def loglog1 (dist/log-logistic-dist 3 2.1))
+(println (dist/describe loglog1 0.75 0.25 2 [0.95]))
+
+;;Student distribution
+(def student1 (dist/student-dist 3))
+(println (dist/describe student1 0.75 0.25 0.5 [0 1]))
+
+(run-tests)
+