Let 𝑋 be a random variable with density

Question

𝑓(𝑥)={𝐶(4𝑥−𝑥^2) , 0≤𝑥≤3

0 , otherwise

where 𝐶∈𝐑 is some constant.

C = 1/9

E[X] = 1.75

Find E[x^2] (the second order moment of 𝑋) = ??

Find var[𝑋], the variance of the random variable 𝑋 = ??

Answer 1

E[x^2] = (1/9)∫0^3 (4x^2 - x^3) dx = (1/9) (16/3 - 1) = 5/9

var[X] = E[x^2] - (E[x])^2 = 5/9 - (1.75)^2 = 0.84375