# binopdf

Binomial probability density function

## Syntax

``y = binopdf(x,n,p)``

## Description

example

````y = binopdf(x,n,p)` computes the binomial probability density function at each of the values in `x` using the corresponding number of trials in `n` and probability of success for each trial in `p`.`x`, `n`, and `p` can be vectors, matrices, or multidimensional arrays of the same size. Alternatively, one or more arguments can be scalars. The `binopdf` function expands scalar inputs to constant arrays with the same dimensions as the other inputs.```

## Examples

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Compute and plot the binomial probability density function for the specified range of integer values, number of trials, and probability of success for each trial.

In one day, a quality assurance inspector tests 200 circuit boards. 2% of the boards have defects. Compute the probability that the inspector will find no defective boards on any given day.

`binopdf(0,200,0.02)`
```ans = 0.0176 ```

Compute the binomial probability density function values at each value from 0 to 200. These values correspond to the probabilities that the inspector will find 0, 1, 2, ..., 200 defective boards on any given day.

```defects = 0:200; y = binopdf(defects,200,.02);```

Plot the resulting binomial probability values.

`plot(defects,y)` Compute the most likely number of defective boards that the inspector finds in a day.

```[x,i] = max(y); defects(i)```
```ans = 4 ```

## Input Arguments

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Values at which to evaluate the binomial pdf, specified as an integer or an array of integers. All values of `x` must belong to the interval ```[0 n]```, where `n` is the number of trials.

Example: `[0,1,3,4]`

Data Types: `single` | `double`

Number of trials, specified as a positive integer or an array of positive integers.

Example: `[10,20,50,100]`

Data Types: `single` | `double`

Probability of success for each trial, specified as a scalar value or an array of scalar values. All values of `p` must belong to the interval `[0 1]`.

Example: `[0.01,0.1,0.5,0.7]`

Data Types: `single` | `double`

## Output Arguments

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Binomial pdf values, returned as a scalar value or array of scalar values. Each element in `y` is the binomial pdf value of the distribution evaluated at the corresponding element in `x`.

Data Types: `single` | `double`

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### Binomial Probability Density Function

The binomial probability density function lets you obtain the probability of observing exactly x successes in n trials, with the probability p of success on a single trial.

The binomial probability density function for a given value x and given pair of parameters n and p is

`$y=f\left(x|n,p\right)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}{q}^{\left(n-x\right)}{I}_{\left(0,1,...,n\right)}\left(x\right)$`

where q = 1 – p. The resulting value y is the probability of observing exactly x successes in n independent trials, where the probability of success in any given trial is p. The indicator function I(0,1,...,n)(x) ensures that x only adopts values of 0, 1, ..., n.

## Alternative Functionality

• `binopdf` is a function specific to binomial distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `pdf`, which supports various probability distributions. To use `pdf`, specify the probability distribution name and its parameters. Alternatively, create a `BinomialDistribution` probability distribution object and pass the object as an input argument. Note that the distribution-specific function `binopdf` is faster than the generic function `pdf`.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.