2024 HCI P2 Q12

The probability distribution function of a discrete random variable, \({X}\), is given as follows:

\({\mathrm{P}(X=x)=\begin{cases} \frac{1}{2}\mathrm{P}\,(X=\left\vert{}x\right\vert{}+1), & \text{if}\hspace{0.5em} x=-2,-1 \\ a, & \text{if}\hspace{0.5em} x=0,1 \\ b, & \text{if}\hspace{0.5em} x=2,3 \\ 0, & \textbf{otherwise} \end{cases}}\)

where \({a}\) and \({b}\) are positive constants.

1. Show that \(b=\frac{1-2a}{3}\).[3]
2. 1. Show that \(\mathrm{E}\left( X \right)=\frac{7}{6}-\frac{4a}{3}\).[3]
   2. Find \(\mathrm{Var}\left( X \right)\) in terms of \(a\)[3]
   3. Find the range of values of \(a\) for which \(\mathrm{Var}\left( X \right)\) exists.[2]

Let \(a=\frac{7}{20}\).

3. A random sample of \(50\) observations of \(X\) is taken. Find the probability that the sum of these observations differs from\(36\) by less than \(5\).[3]