## The Z - Transform

Suppose a random variable
$X$
follows a normal distribution with mean
$\mu$
and standard variance
$\sigma^2$
.
We write
$X \sim N( \mu , \sigma^2)$
.
Just knowing that a random variable follows a normal distribution is not a lot of use if we cannot calculate the probability of
$X$
taking a certain value, or given a probability, finding the correspomnding value of
$X$
.
To be able to do this we use the Z - transform to transform any normal distribution
$N( \mu , \sigma^2 )$
onto the standard normal distribution
$N(0, 1)$
.
The Z - transform is
$Z=\frac{X - \mu}{\sigma}$
. Using the transform means we can use a single table instead of a range of tables for possible values of
$\mu$
and
$\sigma$
.