Answer to Question #119485 in Statistics and Probability for Kyei William Frimpong

Question

A random variable X has the cumulative distribution function given as
F(x) =



0, for x < 1
x
2
2 2x + 2
2
, for 1 ≤ x < 2
1, for x ≥ 2

Answer 1

P(1<x<2) = F(2) - F(1)= (2² -2*(2) + 2)/2 - (1² - 2*(1) + 2)/2= (4-4+2)/2 - (1-2+2)/2 = 2/2 - 1/2= 1/2

So if 1/2 of the pdf lies between 1<x<=2, none of it is x>2, and none of it is below x<1, the remaining 1/2 of the pdf must lie at x=1, so

f(x)=F'(x)={1/2,x=1;x-1,1<x<2;0 - otherwise

E(x)=1/2+∫²₁x(x-1)dx=4/3= 1.333

E(x2)=1/2+∫²₁x²(x-1)dx=23/12=1.916

VAR=|E(x²)-E(x)²|=0.139