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affeldt-aist merged 1 commit into from
Jul 24, 2026
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chore: 📝 Rocqdoc deflist sample #2047

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74 changes: 48 additions & 26 deletions theories/probability_theory/random_variable.v
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Expand Up @@ -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` *)
(* *)
(* *)
(******************************************************************************)
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