Geometric distributions over ℕ #
Define the geometric measure over the natural numbers
Main definitions #
geometricPMFReal
: the functionp n ↦ (1-p) ^ n * p
forn ∈ ℕ
, which is the probability density function of a geometric distribution with success probabilityp ∈ (0,1]
.geometricPMF
:ℝ≥0∞
-valued pmf,geometricPMF p = ENNReal.ofReal (geometricPMFReal p)
.geometricMeasure
: a geometric measure onℕ
, parametrized by its success probabilityp
.
The pmf of the geometric distribution depending on its success probability.
Instances For
theorem
ProbabilityTheory.geometricPMFRealSum
{p : ℝ}
(hp_pos : 0 < p)
(hp_le_one : p ≤ 1)
:
HasSum (fun (n : ℕ) => geometricPMFReal p n) 1
The geometric pmf is measurable.
theorem
ProbabilityTheory.isProbabilityMeasureGeometric
{p : ℝ}
(hp_pos : 0 < p)
(hp_le_one : p ≤ 1)
:
MeasureTheory.IsProbabilityMeasure (geometricMeasure hp_pos hp_le_one)