The random variable $X$ has probability function $\mathrm{P}(X=x)=\left\{ \begin{matrix} kx\,\,\,\,\

Question

The random variable $X$ has probability function $\mathrm{P}(X=x)=\left\{ \begin{matrix} kx\,\,\,\,\,\mathrm{for}\,x=3,4,5, \ k\left( 11-x ight)\,\,\,\mathrm{for}\,x=6,7,8. \\end{matrix} ight.$ where $k$ is a constant. Show that $k=\frac{1}{24}$. [1] Find the exact values of $\mathrm{E}(X)$ and $\mathrm{Var}(X)$. [3] Fifty independent observations of $X$ are taken, using a suitable approximation, estimate the probability that the mean of these observations is at less than $4.9$. [3] Given that $Y=mX-1$ where $m$ is a constant, find $\mathrm{E}({{Y}^{2}})$ in terms of $m$. [3] Given now that $Y=3X-1$, find the largest integer $a$ such that $\mathrm{P}(Y\ge a)\ge 0.5$. [3]

Accepted Answer

(b) $\mathrm{E}(X) = \frac{11}{2}$, $\mathrm{Var}(X) = \frac{29}{12}$ (c) $\mathrm{P}(\bar{X} < 4.9) = 0.00317$ (3 s.f.) (d) $\mathrm{E}(Y^2) = \frac{98}{3}m^2 - 11m + 1$ (e) $a = 17$

2023 TJC P2 Q8

2023 TJC P2 Q8

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