The Gaussian Distribution¶
The Standard Normal Distribution¶
\[\mathcal{N}(0,1)\]
\[f_X(x)=\dfrac{1}{\sqrt{2\pi}}\,\exp\left(-\dfrac{x^2}{2}\right)\]
Expectation and Variance¶
\[\mathbb{E}[X]=0\]
\[\mathrm{var}[X]=1\]
General Normal Random Variables¶
\[f_X(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,\exp\left[-\dfrac{(x-\mu)^2}{2\sigma^2}\right]\]
Expectation and Variance¶
\[\mathbb{E}[X]=\mu\]
\[\mathrm{var}[X]=\sigma^2\]
Linear Functions of a Normal Random Variable¶
Let \(Y=aX+b\) where \(X\sim\mathcal{N}(\mu,\sigma^2)\). Then,
\[\mathbb{E}[Y]=a\mu+b\qquad\text{and}\qquad\mathrm{var}[Y]=a^2\sigma^2.\]
\(Y\) is also a normal random variable, i.e.,
\[Y\sim\mathcal{N}(a\mu,a^2\sigma^2).\]