Gaussian Sampling

Sampling \(x\) from \(N(\mu,\sigma^{2})\) is calculated as:

\[\begin{align*} x\sim N(\mu,\sigma^{2}) \\ x = \mu + \sigma. \epsilon \\ \epsilon \sim N(0,1) \end{align*}\]

This is because:

Mean of \(Y=aX\) is:

\[E(Y)=E(aX) = a.E(X)\]

Whereas, Variance of \(Y\) is:

\[Var(Y) = a^{2}.Var(X)\]

Why?

The variance of a random variable \(X\) is:

\[Var(X) = E(X-E(X))^{2}\]

Suppose we define a new variable by scaling \(X\):

\[Y = aX\]

Then the variance of \(Y\) is:

\[\begin{align*} Var(Y) &= Var(aX) \\ &= E(aX-E(aX))^{2} \\ &= E(a^{2}.E(X-E(X))^{2} \\ &= a^{2}.E(X-E(X))^{2} \\ &= a^{2}.Var(X) \end{align*}\]