Combining Two Normally Distributed Random Variables

Throughout this lesson, we write $N(\mu, \sigma^2)$ for the normal distribution with mean $\mu$ and variance $\sigma^2$ (so the standard deviation is $\sigma$).

A key property of the normal distribution is that it is preserved under linear transformations. If $X \sim N(\mu, \sigma^2)$ and $a, b$ are constants with $a \neq 0,$ then the random variable $aX + b$ is also normally distributed:

$$aX + b \;\sim\; N\!\left(a\mu + b,\; a^2 \sigma^2\right).$$

The mean and variance follow from the standard rules:

• $E[aX+b] = aE[X] + b = a\mu + b,$
• $\operatorname{Var}(aX+b) = a^2 \operatorname{Var}(X) = a^2 \sigma^2.$

Notice that the constant $b$ shifts the mean but does not affect the variance, and the scaling factor $a$ is squared when applied to the variance.

For example, if $X \sim N(30, 25),$ then $Y = 2X - 7$ is normal with

$$\begin{align*} E[Y] &= 2(30) - 7 = 53, \\[3pt] \operatorname{Var}(Y) &= 2^2 \cdot 25 = 100, \end{align*}$$

so $Y \sim N(53, 100).$