<ol class="abc-list"><li><div class="list-item-paragraph">The discrete random variable $X$ takes values $-1$, $0$, $1$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ respectively. The variable $\bar{X}$ is the mean of a random sample of $3$ values of $X$ (i.e. ${{X}_{1}}$, ${{X}_{2}}$ and ${{X}_{3}}$ are independent random variables).</div><div class="list-item-paragraph">Tabulate the probability distribution of $\bar{X}$, and use your values to calculate $\mathrm{Var}\left( {\bar{X}} \right)$.</div><div class="list-item-paragraph">Hence verify that $\mathrm{Var}\left( {\bar{X}} \right)=\frac{1}{3}\mathrm{Var}\left( X \right)$ in this case.</div></li></ol>

The discrete random variable $X$ takes values $-1$, $0$, $1$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ respectively. The variable $\bar{X}$ is the mean of a random sample of $3$ values of $X$ (i.e. ${{X}_{1}}$, ${{X}_{2}}$ and ${{X}_{3}}$ are independent random variables).

Tabulate the probability distribution of $\bar{X}$, and use your values to calculate $\mathrm{Var}\left( {\bar{X}} \right)$.

Hence verify that $\mathrm{Var}\left( {\bar{X}} \right)=\frac{1}{3}\mathrm{Var}\left( X \right)$ in this case.

N1984 P2 Q11

N1984 P2 Q11

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