Two independent normal variables
We showed earlier that a linear function of two independent random variables, \(X\) and \(Y\), with means \(\mu_X\) and \(\mu_Y\) and variances \(\sigma_X^2\) and \(\sigma_Y^2\) has a distribution with mean and variance
\[ \begin {align} E[aX + bY] & = a\mu_X + b\mu_Y \\[0.5em] \Var(aX + bY) & = a^2\sigma_X^2 + b^2\sigma_Y^2 \end {align} \]When these two variables have normal distributions, we can be more precise about the shape of the resulting distribution. (The proof requires methodology that we have not covered yet.)
Linear function of independent normal variables
If \(X\) and \(Y\) are independent random variables,
\[ \begin {align} X \;&\sim\; \NormalDistn(\mu_X,\; \sigma_X^2) \\ Y \;&\sim\; \NormalDistn(\mu_Y,\; \sigma_Y^2) \end {align} \]then
\[ aX + bY \;\sim\; \NormalDistn(a\mu_X + b\mu_Y,\; a^2\sigma_X^2 + b^2\sigma_Y^2) \]Random sample
Applying this to the sum of a random sample of \(n = 2\) values from a \(\NormalDistn(\mu,\; \sigma^2)\) distribution,
\[ X_1 + X_2 \;\sim\; \NormalDistn(2\mu,\; 2\sigma^2) \]We now generalise this result.
Sum of a random sample
If \(\{X_1, X_2, ..., X_n\}\) is a random sample of n values from a \(\NormalDistn(\mu,\; \sigma^2)\) distribution then,
\[ \sum_{i=1}^n {X_i} \;\sim\; \NormalDistn(n\mu,\; n\sigma^2) \]We showed above that
\[ X_1 + X_2 \;\sim\; \NormalDistn(2\mu,\; 2\sigma^2) \]Now
\[ \begin{align} X_1 + X_2 + X_3 \;=\;(X_1+ X_2) + X_3 \;&\sim\; \NormalDistn(\mu_{X_1 + X_2} + \mu_{X_3},\; \sigma_{X_1 + X_2}^2 + \sigma_{X_3}^2) \\[0.4em] &\sim\; \NormalDistn(3\mu,\; 3\sigma^2) \end{align} \]In general, the result for a sample of size \(n\) can be proved from that for a sample of \((n-1)\) values, completing a proof by induction.
In a similar way, the mean of a random sample from a normal distribution is also normally distributed.
Mean of a random sample
If \(\{X_1, X_2, ..., X_n\}\) is a random sample of n values from a \(\NormalDistn(\mu,\; \sigma^2)\) distribution then,
\[ \overline{X} \;\sim\; \NormalDistn\left(\mu,\; \frac {\sigma^2}{n}\right) \]