A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that \(P(X

Question

A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that $P(X=5)=p$ and $P(X=-3)=1-p$, where $0\le p\le 1$. Show that $E(X)=-3+8p$ and $\operatorname{Var}(X)=64p(1-p)$.[3] Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.[2] Find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.[3]

Accepted Answer

(i) $E(X)=-3+8p$, $\operatorname{Var}(X)=64p(1-p)$. (ii) Maximum $E(X)=5$ at $p=1$; minimum $E(X)=-3$ at $p=0$. (iii) Maximum variance $=16$ at $p=\dfrac12$; minimum variance $=0$ at $p=0$ or $p=1$.

Discrete Random Variables (DRV) - Constructing and Using Discrete Probability Distributions

Discrete Random Variables (DRV) - Constructing and Using Discrete Probability Distributions

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