Let X be a random variable with cumulative distribution func

Let X be a random variable with cumulative distribution function F(a) = {0 if a < 1; 0.4 if 1 a < 2; 0.7 if 2 a < 3; 0.9 if 3 a < 4; 1 if a 4}. Is X a discrete or continuous random variable? Justify your answer

Solution

From CDF we have probabilites

P(X=1)=F(1)=0.4

P(X=2)=F(2)-F(1)=0.7-0.4=0.3

P(X=3)=F(3)-F(2)=0.9-0.7=0.2

P(X=4)=F(4)-F(3)=1-0.9=0.1

Since probability defined at particular points so X is a discrete random variable.