(a) Let X denote the number of hours a Class 12 student studies during a randomly selected school day. The probability that X can take the values…

Let X denote the number of hours a Class $\displaystyle 12$ student studies during a randomly selected school day. The probability that X can take the values $\displaystyle \mathrm{x}_{\mathrm{i}}$, for an unknown constant 'k' :

$$\mathrm{P}(\mathrm{X}=\mathrm{k})=\left\{\begin{array}{ccc}0.1 & \text { if } & \mathrm{x}_{\mathrm{i}}=0 \\ \mathrm{kx}_{\mathrm{i}} & \text { if } & \mathrm{x}_{\mathrm{i}}=1 \text { or } 2 \\ \mathrm{k}\left(5-\mathrm{x}_{\mathrm{i}}\right) & \text { if } & \mathrm{x}_{\mathrm{i}}=3 \text { or } 4\end{array}\right.$$

Find the value of k.

Determine the probability that the student studied for at least $\displaystyle 2$ hours.

Determine the probability that the student studied for at most $\displaystyle 2$ hours.

A river near a small town floods and overflows twice in every $\displaystyle 10$ years on an average. Assuming that the Poisson distribution is appropriate, what is the mean expectation? Also, calculate the probability of $\displaystyle 3$ or less overflows and floods in a $\displaystyle 10$-year interval. [Given $\displaystyle \mathrm{e}^{-2}=0.13534$]