Question

Question 5: Look again at the formula for a standardized deviate, and explain why values drawn from a normal popula where -0 and σ-1 can be considered as standardized normal deviates. We can rewrite the formula for the variables shown in the distribution as:

Answer 1

Let Y is normally distributed with,

$Mean = \mu$

$Standard deviation = \sigma$

Now let us transform the Y variable by deducting Mean and then dividing the difference by Standard deviation.

$Z = \frac{Y-\mu}{\sigma}$

Let us obtain the Mean and standard deviation of Z

$E(Z) = E\left ( \frac{Y-\mu}{\sigma} \right ) = \frac{E(Y)-\mu}{\sigma} = \frac{\mu-\mu}{\sigma} = 0$

$Var(Z) = Var\left ( \frac{Y-\mu}{\sigma} \right ) = \frac{Var(Y)+0}{\sigma^2} = \frac{\sigma^2}{\sigma^2} = 1$

$Standard\ deviation = \sqrt{Var(Y) } = \sqrt{1} = 1$

So, we can see that the Mean = 0 and Standard deviation = 1 for Z which is termed as Standard Normal variate.