# Tag Archives: Standard Normal Distribution

## Normal Approximation

If a random variable $Y$ is normal, you can map it to a standard normal distribution $X$ (useful for finding probabilities in the standard normal table) by the following relationship:

$Y = \mu_y + \sigma_yX$

Example 1:  $Y$ is normal.  $E[Y] = 100$ and $Var(Y) = 49$  Then

$\begin{array}{rl} P(Y \leq 111.515) &= P(X \leq \frac{111.515 - 100}{\sqrt{49}}) \\ &= P(X \leq 1.645) \\ &= 0.95 \end{array}$

Example 2:  $Y$ has the same distribution as example 1.  Then $P(Y \leq y) = 0.9$ implies

$P(X \leq \frac{y - 100}{\sqrt{49}}) = 0.9$

Which implies:

$\frac{y - 100}{\sqrt{49}} = 0.8159$

Hence $y = 105.7113$.

With regard to Central Limit Theorem:

By the Central Limit Theorem, the distribution of a sum of iid random variables converges to a normal distribution as the number of iid random variables increases.  This means that if the number of iid random variables is sufficiently large, we can get approximate probabilities by using a normal distribution approximation.