Question

A discrete random variable X has the following probability distribution:
x 1 2 3 4
P(X=x) 1/3 1/3 k 1/4
where k is a constant.
(a) Find the value of k.
(b) Find P(X ≤3).

Answer 1

(a):

$\sum P(x)=1$

So,

$\frac {1}{3} +\frac {1}{3} +k+\frac {1}{4} =1$

$\frac {4+4+3}{12} +k=1$

$\frac {11}{12} +k=1$

$k=1-\frac {11}{12}$

$k=\frac {1}{12}$

(b):

$P(X\le3)=P(X=1)+P(X=2)+P(X=3)$

$P(X\le3)=\frac{1}{3}+\frac{1}{3}+\frac{1}{12}$

$P(X\le3)=\frac{4+4+1}{12}$

$P(X\le3)=\frac{9}{12}$

$P(X\le3)=\frac{3}{4}$