Q.The p.m.f. of a r.v. X is given by P (X = x) = (5 Cx)/2⁵, for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise. Then show that P (X ≤ 2) = P (X ≥ 3).

For a discrete random variable $X$ with possible values $x_1, x_2, x_3, \ldots$ occurring with probabilities $p_1, p_2, p_3, \ldots$, where $p_i = P[X = x_i]$, the rule that assigns each $x_i$ its probability $p_i$ is called the probability mass function of $X$, provided every $p_i \ge 0$ and the probabilities add up to 1, i.e. $\sum p_i = 1$. Listing the pairs $(x_i, p_i)$, often in a two-row table, gives the probability distribution of $X$. Every possible value must genuinely occur with positive probability, and because the values of $X$ between t …