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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```