The random variable $X$ is such that $X\sim$ $N\left( m,3.7 \right)$ and ${{X}_{1}}$ an

Question

The random variable $X$ is such that $X\sim$ $N\left( m,3.7 \right)$ and ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Write down an expression for the distribution of the random variable $3{{X}_{1}}-{{X}_{2}}$ in terms of $m$ and find the greatest value of $\mathrm{P}\left( -6<3{{X}_{1}}-{{X}_{2}}<0 \right)$.[4] The random variable $\bar{X}$ is the mean of $n$ independent observations of $X$ and $\mathrm{P}\left( \bar{X}>1 \right)\le 0.01$. Find an inequality satisfied by $m$ and $n$.[3]

Accepted Answer

(a) ${3X_1 - X_2 \sim N(2m, 37)}$ ; greatest ${\mathrm{P}(-6 < 3X_1 - X_2 < 0) = 0.378}$ ($3$ s.f.) (b) ${(1-m)\sqrt{n} \geq 4.47}$

2023 VJC P2 Q8

2023 VJC P2 Q8

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