$X$ is a random variable such that $\mathrm{E}(X)=4$ and $\mathrm{E}[(X-3)^2]=40$. Find t

Question

$X$ is a random variable such that $\mathrm{E}(X)=4$ and $\mathrm{E}[(X-3)^2]=40$. Find the value of $\mathrm{Var}(X)$.[3] Let $Y$ be a discrete random variable. Let $\mathrm{f}$ be a function which is defined for all values that $Y$ can take. Then $\mathrm{f}(Y)$ is also a random variable, and its expectation is given by $$\mathrm{E}[\mathrm{f}(Y)] = \sum_{y} [\mathrm{f}(y) \times \mathrm{P}(Y=y)].$$ Suppose now that $Y \sim \mathrm{B}(3, \frac{1}{3})$ and $\mathrm{f}(t) = \frac{1}{t+1}$. Find the value of $\mathrm{E}[\mathrm{f}(Y)]$. [3] $W$ is a random variable with the following probability distribution: $w$$-3$$-1$$1$$2$ $\mathrm{P}(W=w)$$p$$q$$0.15$$r$ It is given that $\mathrm{E}(W)=-0.2$ and $\mathrm{E}(W^3)=-3.2$. Find the values of $p$, $q$ and $r$.[4]

Accepted Answer

(a) $39$. (b) $\frac{65}{108}$. (c) $p=0.2$, $q=0.35$, $r=0.3$.

2025 EJC P2 Q8

2025 EJC P2 Q8

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