The probability distribution of a discrete random variable $X$ is given by $x$$0$$2$$4$ \(

Question

The probability distribution of a discrete random variable $X$ is given by $x$$0$$2$$4$ $P(X=x)$$p$$p$$1-2p$ State the range of possible values of $p$.[1] Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.[2] Show that $\operatorname{Var}(X)=4p(5-9p)$.[2] Hence find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.[3]

Accepted Answer

(i) $0\le p\le\dfrac12$. (ii) Maximum $E(X)=4$ at $p=0$; minimum $E(X)=1$ at $p=\dfrac12$. (iii) $\operatorname{Var}(X)=4p(5-9p)$. (iv) Maximum variance $=\dfrac{25}{9}$ at $p=\dfrac{5}{18}$; minimum variance $=0$ at $p=0$.

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

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

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