The random variable $X$ is normally distributed with mean $20$ and variance $\sigma^{2}$. Give

Question

The random variable $X$ is normally distributed with mean $20$ and variance $\sigma^{2}$. Given that $\mathrm{P}(12k)=0.42065$, show that $\sigma =8$, correct to the nearest whole number, and find the value of the constant $k$.[4] $X$ 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 $Y$. Another random variable $W$, where $W$ and $Y$ are independent, is normally distributed with mean $25$ and variance $16$. Using $\sigma =8$, find $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$, and hence find $\mathrm{P}\left( W>2Y \right)$.[5]

Accepted Answer

(i) $\sigma=8,\ k=21.6$ (ii) $E(Y)=12,\ \operatorname{Var}(Y)=32,\ P(W>2Y)=0.533$

Advanced Operations, Probability Modeling, and Parameter Estimation

Advanced Operations, Probability Modeling, and Parameter Estimation

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