Geometric Distribution

The Geometric Distribution is a probability mass function that models the probability of the first success occurring for a given trial, such as:

Example

• The first student owning an android out of $n$
• The first person to sit in a seat in a library
• The first dog that is trained in a group

The geometric distribution is a special case of the binomial distribution with $n$ set to $1$.

It is an example of a Memoryless Distribution.

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Formula:

$Geo(x)=(1-p{)}^{k-1}p$

Where

• $p$ is the probability of success
• $k$ is the count of trials before the value is expected

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Cumulative Distribution Function

The cumulative distribution function for the geometric distribution is: $P(X\le k)=1-(1-p{)}^k$

Properties

Expected Value

The expected value of the geometric distribution is: $\frac{1}{p}$

Variance

The variance of the geometric distribution is: $\frac{1-p}{p^2}$