<p>The random variable $X$ has probability function $\mathrm{P}(X=x)=\left\{ \begin{matrix} kx\,\,\,\,\,\mathrm{for}\,x=3,4,5, \\ k\left( 11-x \right)\,\,\,\mathrm{for}\,x=6,7,8. \\\end{matrix} \right.$</p>
<p>where $k$ is a constant.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $k=\frac{1}{24}$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the exact values of $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Fifty independent observations of $X$ are taken, using a suitable approximation, estimate the probability that the mean of these observations is at less than $4.9$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Given that $Y=mX-1$ where $m$ is a constant, find $\mathrm{E}({{Y}^{2}})$ in terms of $m$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Given now that $Y=3X-1$, find the largest integer $a$ such that $\mathrm{P}(Y\ge a)\ge 0.5$.</span></div><p style="text-align: right;">[3]</p></li></ol>

The random variable $X$ has probability function $\mathrm{P}(X=x)=\left\{ \begin{matrix} kx\,\,\,\,\,\mathrm{for}\,x=3,4,5, \\ k\left( 11-x \right)\,\,\,\mathrm{for}\,x=6,7,8. \\\end{matrix} \right.$

Find the exact values of $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

Fifty independent observations of $X$ are taken, using a suitable approximation, estimate the probability that the mean of these observations is at less than $4.9$.

Given that $Y=mX-1$ where $m$ is a constant, find $\mathrm{E}({{Y}^{2}})$ in terms of $m$.

Given now that $Y=3X-1$, find the largest integer $a$ such that $\mathrm{P}(Y\ge a)\ge 0.5$.

2023 TJC P2 Q8

2023 TJC P2 Q8

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