The random variable $X$ is normally distributed with mean $135$ variance ${{\sigma }^{2}}$. Given th

Question

The random variable $X$ is normally distributed with mean $135$ variance ${{\sigma }^{2}}$. Given that $\mathrm{P}\left( 127a \right)=0.3865$, 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 $Y$. Find $\mathrm{E}\left( Y \right)$ and $\mathrm{Var}\left( Y \right)$.

Accepted Answer

(i) $\sigma = 10.68$ (2 d.p.), $a = 138$; (ii) $\text{E}(Y) = 69.5$, $\text{Var}(Y) = 57.1$

Worked Example 6

Worked Example 6

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