The random variable $X$ has the distribution $ \mathrm{N}(50, 8^2)$. Given that $X_1$ and

Question

The random variable $X$ has the distribution $ \mathrm{N}(50, 8^2)$. Given that $X_1$ and $X_2$ are two independent observations of $X$, find $P(X_1 + X_2 > 120)$,[2] $P(X_1 > X_2 + 15)$.[3] The random variable $Y$ is related to $X$ by the formula $Y = aX + b$, where $a$ and $b$ are constants with $a > 0$. Given that $ \mathrm{P}(Y < 74) = \mathrm{P}(Y > 146) = 0.0668$, find the values of $ \mathrm{E}(Y)$ and $ \mathrm{Var}(Y)$, and hence find the values of $a$ and $b$.[7]

Accepted Answer

(i) $0.0385$ (ii) $0.0924$ (iii) $\mathrm E(Y)=110$, $\mathrm{Var}(Y)=576$, $a=3$, $b=-40$

RI Normal Distribution Tutorial Q4

RI Normal Distribution Tutorial Q4

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