diff --git a/theories/probability_theory/random_variable.v b/theories/probability_theory/random_variable.v index 99dbb9f7b4..6447db2ec3 100644 --- a/theories/probability_theory/random_variable.v +++ b/theories/probability_theory/random_variable.v @@ -19,32 +19,54 @@ From mathcomp Require Import lebesgue_measure lebesgue_integral hoelder. (* of `Lfun P n` hypotheses instead of the `integrable` predicate from *) (* `lebesgue_integral.v`. *) (* *) -(* ``` *) -(* {RV P >-> T'} == random variable: a measurable function to the *) -(* measurableType T' from the measured space *) -(* characterized by the probability P *) -(* distribution P X == measure image of the probability measure P by the *) -(* random variable X : {RV P -> T'} *) -(* P as type probability T R with T of type *) -(* measurableType. *) -(* Declared as an instance of probability measure. *) -(* 'E_P[X] == expectation of the real measurable function X *) -(* covariance X Y == covariance between real random variable X and Y *) -(* 'V_P[X] == variance of the real random variable X *) -(* 'M_P X == moment generating function of the random variable X *) -(* with sample space corresponding to the probability *) -(* measure P *) -(* {dmfun T >-> R} == type of discrete real-valued measurable functions *) -(* {dRV P >-> R} == real-valued discrete random variable *) -(* dRV_dom X == domain of the discrete random variable X *) -(* dRV_enum X == bijection between the domain and the range of X *) -(* pmf X r := fine (P (X @^-1` [set r])) *) -(* cdf X r == cumulative distribution function of X *) -(* := distribution P X `]-oo, r] *) -(* ccdf X r == complementary cumulative distribution function of X *) -(* := distribution P X `]r, +oo[ *) -(* enum_prob X k == probability of the kth value in the range of X *) -(* ``` *) +(* `{RV P >-> T'}` *) +(* ~ random variable: a measurable function to the `measurableType T'` *) +(* from the measured space characterized by the probability `P` *) +(* *) +(* `distribution P X` *) +(* ~ measure image of the probability measure `P` by the random variable *) +(* `X : {RV P -> T'}` *) +(* ~ `P` has type `probability T R` with `T` of type `measurableType`. *) +(* ~ Declared as an instance of probability measure. *) +(* *) +(* `'E_P[X]` *) +(* ~ expectation of the real measurable function `X` *) +(* *) +(* `covariance X Y` *) +(* ~ covariance between real random variable `X` and `Y` *) +(* *) +(* `'V_P[X]` *) +(* ~ variance of the real random variable `X` *) +(* *) +(* `'M_P X` *) +(* ~ moment generating function of the random variable `X` with sample *) +(* space corresponding to the probability measure `P` *) +(* *) +(* `{dmfun T >-> R}` *) +(* ~ type of discrete real-valued measurable functions *) +(* *) +(* `{dRV P >-> R}` *) +(* ~ real-valued discrete random variable *) +(* *) +(* `dRV_dom X` *) +(* ~ domain of the discrete random variable `X` *) +(* *) +(* `dRV_enum X` *) +(* ~ bijection between the domain and the range of `X` *) +(* *) +(* `pmf X r` *) +(* ~ ``fine (P (X @^-1` [set r]))`` *) +(* *) +(* `cdf X r` *) +(* ~ cumulative distribution function of `X` *) +(* ~ ``:= distribution P X `]-oo, r]`` *) +(* *) +(* `ccdf X r` *) +(* ~ complementary cumulative distribution function of `X` *) +(* ~ := ``distribution P X `]r, +oo[`` *) +(* *) +(* `enum_prob X k` *) +(* ~ probability of the kth value in the range of `X` *) (* *) (* *) (******************************************************************************)