Question

The random variable $X$ has a probability distribution $P\left(X\right)$ of the following form, where $k$ is some number :
$P\left(X\right)=\{k,ifx=02k,ifx=13k,ifx=20,otherwise\}$
(a) Determine the value of $k$
(b) Find $P(X<2),P(X\le 2),P(X\ge 2)$.

Answer 1

It is known that the sum of probabilities of a probability distribution of random variables is one.
$\therefore k+2k+3k+0=1$
$\Rightarrow 6k=1$
$\Rightarrow k=\frac{1}{6}$
(b) $P(X<2)=P(X=0)+P(X=1)$
$=k+2k$
$=3k$
$=\frac{3}{6}$
$=\frac{1}{2}$
$P(X\le 2)=P(X=0)+P(X=1)+P(X=2)$
$=k+2k+3k$
$=6k$
$=\frac{6}{6}$
$=1$
$P(X\ge 2)=P(X=2)+P(X>2)$
$=3k+0$
$=3k$
$=\frac{3}{6}$
$=\frac{1}{2}$