(a) A random variable X has the following probability function : | $X…

Previous year question for Engineering Mathematics-III.

Question

(a) A random variable X has the following probability function :

| $X=x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |

|---|---|---|---|---|---|---|---|---|---|

| $P(x)$ | $k$ | $3k$ | $5k$ | $7k$ | $9k$ | $11k$ | $13k$ | $15k$ | $17k$ |

(i) Find the value of $k$.

(ii) Evaluate $P(X \ge 6)$, $P(3 < X < 5)$.

(iii) Calculate the minimum value of $k$ such that $P(X \le 3) \ge 0.5$.

(b) Find the variance of the number of divisors of an integer from 1 through 20 chosen at random.