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  
\[\sigma\]
.