Probability density function

Since \(Y = a + bX\) is a monotonic transformation, we can apply the earlier general results to find the pdf of \(Y\). Writing

\[ y = g(x) = a + bx \qquad x = h(y) = \frac{y-a}{b} \spaced{and} h'(y) = \frac 1 b \]

The random variable \(Y\) has pdf

\[ f_Y(y) \;\;=\;\; f_X\left(h(y)\right) \times h'(y) \;\;=\;\; \frac 1 {\left|b\right|} f_X\left(\frac{y-a}{b}\right) \]

We now apply this to linear transformations of a normal random variable.

Linear transformation of a normal variable

If \(a\) and \(b\) are constants and \(X \sim \NormalDistn(\mu, \sigma^2)\), the random variable \(Y = (a + bX)\) also has a normal distribution

\[ Y \;\;\sim\;\; \NormalDistn(a + b\mu,\; b^2 \sigma^2) \]

(Proved in full version)

In particular, this result provides the distribution of a z-score.

Distribution of z-scores

If \(X \sim \NormalDistn(\mu, \sigma^2)\), the random variable \(Z = \dfrac {X-\mu} {\sigma} \) has a normal distribution with zero mean and standard deviation one,

\[ Z \sim \NormalDistn(0, 1) \]

(Proved in full version)

Probabilities about z-scores can be found using computer software or tables.