Geometric Distribution
A geometric distribution is a Discrete Distribution of Random Variables
Assume we run a series of Bernoulli Trials where the probability of seing the event $A$ is $p$, and, therefore, the probability of not seing $A$ is $q = 1 - p$
The trials stop once $A$ occures, i.e. if $A$ occures at $k$-th trial, it didn’t occur in previous $k -1$ trials
Random Variable $X$ is the number of trials we should run until we see $A$
- the distribution of $X$ is called Geomentric
Formally, Geometric Distribution describes the waiting time until a success for indepented and identically distributed Bernoulli Random Variables
Typical questions:
- How long should we flip a coin until we get head?
- How many times we roll a dice until we get 1?
Cumulative Distribution Function
Suppose the event did not occur in the $(k-1)$-th trial, but occurred in the $k$-th trial. Then by the multiplication theorem for independent events we have the following distribution function:
$P(X = k) = q^{k - 1} p$
Thus, for each $k = 0, 1, 2, …$ we obtain a geometric progression, where $p$ is the first term and $q$ is the common ratio:
$p, qp, q^2 p, …, q^{k - 1} p, …$
Moments
- $E[X] = \cfrac{p}{1 - p}$
- $\text{Var}[X] = \cfrac{q}{p^2}$
See Also
- Hypergeometric Distribution
- Negative Binomial Distribution - general case of Geometric distribution
Sources
- Gmurman V.E., Probability Theory and Mathematical Statistics – 9th edition. Moscow: Vyssh. shk., 2003.
- OpenIntro Statistics (book)