Mean and Variance of the Normal Distribution

Identify the mean and variance of a normal random variable from the notation N(μ, σ²) and from its probability density function, and write the PDF given the mean and variance.

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Tutorial

Introduction

If a random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2,$ we write

X \sim N(\mu, \sigma^2).

This means

E[X] = \mu, \qquad \mathrm{Var}(X) = \sigma^2.

The standard deviation is the square root of the variance:

\sigma = \sqrt{\sigma^2}.

Note: The second slot inside the parentheses of $N(\mu, \sigma^2)$ is the variance, not the standard deviation.

For instance, if $X \sim N(3, 25),$ then

\mu = 3, \qquad \sigma^2 = 25, \qquad \sigma = \sqrt{25} = 5.