Geometric distribution: setup and mean.

medium

Answer

$X ~ Geometric(p): number of trials until the first success in i.i.d.$
$Bernoulli(p).$
$PMF: P(X=k) = (1-p)^(k-1) * p.$
$\operatorname{Var}(X)\; = \;(1 - p) / p^{2}$
$P(X\; > \;n + k\; \mid \;X\; > \;n)\; = \;P(X\; > \;k)$
$Uses: number of attempts until conversion, retries until failure.$
$Note: two conventions — some define it as failures before first success (E = (1-p)/p).$

Check yourself — multiple choice

E = p
Trials until first success; E[X] = 1/p, $\operatorname{Var}\; = \;(1 - p) / p^{2}$ ; memoryless
Same as Poisson
No mean

Geometric: E = 1/p; memoryless — 'starts over' after every failure.

#distributions#probability

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