Geometric Distribution

Jul 24, 2024 12:00 AM

Aug 18, 2024 6:51 PM

Geometric Distribution

Definitions

a series of independent trials
each trial can be a failure or success, and the probability of success is the same for each trial
how many trials are needed in order to get the first successful outcome

P (X = r) = p q^{r - 1}

where

$X$ is the number of trials needed to get the first successful outcome
$p$ is the probability of success
$q = 1 - p$

Inequalities

$P (X > r) = q^{r}$
- The probability you’ll need more than $r$ trials to get your first success
$P (X \leq r) = 1 - q^{r}$
- The probability you’ll need $r$ trials or less to get your first success

If a variable $X$ follows a geometric distribution where the probability of success in a trial is $p$ , this can be written as

$X \sim G e o (p)$

Expectation

E (X) = \frac{1}{p}

Variance

V a r (X) = \frac{q}{p^{2}}

Sources: 1

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