2017 IJC BT2 Q7
The discrete random variable $X$ has probability function
$\text{P}\left( X=x \right)=\left\{ \begin{matrix}
k\left( \frac{5}{2}-x \right),\,\,\,x=0,2, \\
k\left( x-2 \right),\,\,\,\,\,\,\,\,\,\,x=3, \\
0,\text{otherwise,} \\
\end{matrix} \right.$
where $k$ is a positive constant.
(i)
Show that $k=\frac{1}{4}$.
[1]
(ii)
Find $\text{E}\left( X \right)$ and show that $\text{Var}\left( X \right)=\frac{7}{4}$.
[3]
(iii)
A discrete random variable $Y$ has $\text{E}\left( Y \right)=2$ and $\text{Var}\left( Y \right)=\frac{24}{5}$. Find the values of $\text{E}\left( 2X-3Y \right)$ and $\text{Var}\left( 2X-3Y \right)$, stating any assumption necessary for the calculations to be valid.
[5]
Given that ${{X}_{1}}$ and ${{X}_{2}}$ are two independent observations of $X$,
(iv)
find the probability distribution of $\left| {{X}_{1}}-{{X}_{2}} \right|$,
[3]
(v)
find the probability that the sum of ${{X}_{1}}$ and ${{X}_{2}}$ exceeds $4$, given that the sum is at least $3$.
[3]
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