$X$ is a normal random variable with mean $1$ and variance $1$, and $Y$ is also a normal ran

Question

$X$ is a normal random variable with mean $1$ and variance $1$, and $Y$ is also a normal random variable with mean $\mu $ and variance $2$. Given that $\mathrm{P}\left( 0\le X+Y<3 \right)>0.44$, find the range of possible values of $\mu $.[3] Given instead that $\mu =10$ and the probability that the sum of $n$ independent observations of $X$ exceeds $2Y$ by at least $10$ is less than $\mathrm{0}\mathrm{.03}$, find the largest possible value of $n$.[4] State an assumption needed for the above calculations to be valid.[1]

Accepted Answer

(i) ${-1.10 < \mu < 2.10}$ (ii) ${20}$ (iii) We need to assume that ${X}$ and ${Y}$ are independent.

2023 EJC P2 Q7

2023 EJC P2 Q7

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