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Definition: Normal Random Variables
A continuous random variable $X $ is said to be $normal $ or $Gaussian $ with the following PDF:
$$f_X(x) = \frac{1}{\sqrt{2\pi}{ \sigma }}e^{-\frac{(x-{ \mu })^2}{2{ \sigma }^2}} $$
Where $\mu$ and $\sigma $ are two scalar parameters characterizing the PDF. It can be verified that, for $X $:
$$E[X] = \mu, \var{X} = \sigma^2 $$
Its CDF is denoted by $\Phi $:
$$\Phi(y) = \b{P}(Y \leq y) = \b{P}(Y < y) = \int_{ -\infty }^{ y } \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}t^2} \d t $$
Remarks: Normality is Preserved by Linear Transformations
If $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2 $ , and if a \neq 0 , $b$ are scalars, then the random variable
$$Y = aX + b $$
is also normal, with mean and variance
$$E[Y] = a \mu + b , var(Y) = a^2 \sigma^2 $$
Definition: Standard Normal Random Variable
A normal random variable $Y $ with zero mean and unit variance is said to be a standard normal. i.e.
$$\mu = 0, \sigma = 1 $$ The standard normal random variable’s PDF is hence:
$$f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} $$