Worked Example 6
The random variable $X$ is normally distributed with mean $135$ variance ${{\sigma }^{2}}$.
(i)
Given that $\text{P}\left( 127a \right)=0.3865$, show that $\sigma =10.68$, correct to the $2$ decimal places, and find the value of the constant $a$.
$X$ is related to a normal random variable $Y$ by formula $X={{Y}_{1}}+{{Y}_{2}}-4$, where ${{Y}_{1}}$ and ${{Y}_{2}}$ are two independent observations of $Y$.
(ii)
Find $\text{E}\left( Y \right)$ and $\text{Var}\left( Y \right)$.
