Question

The random variable x has a probability distribution $P\left(x\right)$ of the following form where $k$ is some number:
$P\left(x\right)=\{\begin{array}{cc}k& ifx=0\\ 2k& ifx=1\\ 3k& ifx=2\\ 0& otherwise\end{array}$
Determine the value of $k$ and $P(X\le 2).$

Answer 1

${\sum }_{x=0}^{\infty }P\left(x\right)=1$

$P\left(1\right)+P\left(2\right)+......=1$

$k+2k+3k=1$

$k=\frac{1}{6}$

$P(x\le 2)=P\left(0\right)+P\left(1\right)+P\left(2\right)=k+2k+3k=6k=1$