The random variable $X$ has a normal distribution with mean $\mu $ and standard deviation $2$.

Question

The random variable $X$ has a normal distribution with mean $\mu $ and standard deviation $2$. Given that $\mathrm{P}\left( X\le 1 \right)=\mathrm{0}\mathrm{.1587}$, find the value of $\mu $, giving your answer to the nearest integer.[2] Given further that ${\mathrm{P}(X > 3k) = \mathrm{P}(X < 9k + 4)}$ and using the answer obtained in part (i), find the value of ${k}$. [1] The random variable $Y$ is related to $X$ by the formula $Y=10-X$. Find $P\left( \overline{Y}>6 \right)$, where $\overline{Y}$ is the mean of two independent observations of $Y$.[3]

Accepted Answer

(i) $\mu = 3$ (ii) $k = \frac{1}{6}$ (iii) $0.760$

2018 MI P2 Q8

2018 MI P2 Q8

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