# Documentation

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# normpdf

Normal probability density function

## Syntax

```Y = normpdf(X,mu,sigma) Y = normpdf(X) Y = normpdf(X,mu) ```

## Description

`Y = normpdf(X,mu,sigma)` computes the pdf at each of the values in `X` using the normal distribution with mean `mu` and standard deviation `sigma`. `X`, `mu`, and `sigma` 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 inputs. The parameters in `sigma` must be positive.

The normal pdf is

`$y=f\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-{\left(x-\mu \right)}^{2}}{2{\sigma }^{2}}}$`

The likelihood function is the pdf viewed as a function of the parameters. Maximum likelihood estimators (MLEs) are the values of the parameters that maximize the likelihood function for a fixed value of `x`.

The standard normal distribution has µ = 0 and σ = 1.

If x is standard normal, then xσ + µ is also normal with mean µ and standard deviation σ. Conversely, if y is normal with mean µ and standard deviation σ, then x = (yµ) / σ is standard normal.

`Y = normpdf(X)` uses the standard normal distribution (`mu = 0`, `sigma = 1`).

`Y = normpdf(X,mu)` uses the normal distribution with unit standard deviation (`sigma = 1`).

## Examples

```mu = [0:0.1:2]; [y i] = max(normpdf(1.5,mu,1)); MLE = mu(i) MLE = 1.5000```