Documentation

# geopdf

Geometric probability density function

## Syntax

```y = geopdf(x,p) ```

## Description

`y = geopdf(x,p)` returns the probability density function (pdf) of the geometric distribution at each value in `x` using the corresponding probabilities in `p`. `x` and `p` can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other input. The parameters in `p` must lie on the interval `[0,1]`.

## Examples

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Suppose you toss a fair coin repeatedly, and a "success" occurs when the coin lands with heads facing up. What is the probability of observing exactly three tails ("failures") before tossing a heads?

To solve, determine the value of the probability density function (pdf) for the geometric distribution at `x` equal to `3`. The probability of success (tossing a heads) `p` in any given trial is `0.5`.

```x = 3; p = 0.5; y = geopdf(x,p)```
```y = 0.0625 ```

The returned value of `y` indicates that the probability of observing exactly three tails before tossing a heads is `0.0625`.

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### Geometric Distribution pdf

The probability distribution function (pdf) of the geometric distribution is

`$y=f\left(x|p\right)=p{\left(1-p\right)}^{x}\text{ };\text{ }x=0,1,2,\dots \text{\hspace{0.17em}},$`

where p is the probability of success, and x is the number of failures before the first success. The result y is the probability of observing exactly x trials before a success, when the probability of success in any given trial is p. For discrete distributions, the probability distribution function is also known as the probability mass function (pmf).