The random variable $X$ has a probability distribution $P(X)$ of the following form, where $k$ is some number:

$P(X) = \left\{ {\begin{array}{llllllllllllllllllll}{k,{\rm{ if }}X = 0}\\{2k,{\rm{ if }}X = 1}\\{3k,{\rm{ if }}X = 2}\\{0,{\rm{ other wise }}}\end{array}} \right.$

(a) Determine the value of $k$ .

(b) Find $P(X < 2),P(X \le 2),P(X \ge 2)$

.: The probability distribution of $X$ is :

(a) Since, $\sum\limits_{i = 1}^n {{p_i}} = 1$
$\therefore \quad k + 2k + 3k = 1 \Rightarrow 6k = 1 \Rightarrow k = \cfrac{1}{6}$

(b) (i) $P(X < 2) = P(X = 0) + P(X = 1)$
$= k + 2k = 3k = 3\left( {\cfrac{1}{6}} \right) = \cfrac{1}{2}$ (By (a))

(ii) $P(X \le 2) = P(X = 0) + P(X = 1) + P(X = 2)$

$= k + 2k + 3k = 6k = 6\left( {\cfrac{1}{6}} \right) = 1$ (By (a))

(iii) $P(X \ge 2) = P(2) = 3k = 3\left( {\cfrac{1}{6}} \right) = \cfrac{1}{2}$

(By (a))