A computer can give independent observations of a random variable $X$ with probability distribution

Question

A computer can give independent observations of a random variable $X$ with probability distribution given by $\mathrm{P}\left( X=0 \right)=\frac{3}{4}$ and $\mathrm{P}\left( X=2 \right)=\frac{1}{4}$. It is programmed to output a value for the random variable $Y$ defined by $Y={{X}_{1}}+{{X}_{2}}$, where ${{X}_{1}}$ and ${{X}_{2}}$ are independent observations of $X$. Tabulate the probability distribution of $Y$, show that $E\left( Y \right)=1$. [3] The random variable $T$ is defined by $T={{Y}^{2}}$. Find $\mathrm{E}\left( T \right)$ and show that $\mathrm{Var}\left( T \right)=\frac{63}{4}$. [3] The computer is programmed to produce a large number $n$ of independent values of $T$ and to calculate the mean $M$ of these values. Find the smallest value of $n$ such that $\mathrm{P}\left( M0.99$. [4]

Accepted Answer

Smallest $n = 341$

N1998 P2 Q7

N1998 P2 Q7

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