The expected value of a random variable X uniformly distributed X ∼ Unif(a,b) is:
E[X] = ∫ₐᵇ xf(x) dx = ∫ₐᵇ {x}/{b-a}dx = {1}/{b-a} [ {1}/{2}x² ]ₐᵇ = {(b²-a²) }/{2(b-a) } = {b+a}/{2}
Now, assume that the random variable is uniformly distributed X ∼ Unif(-1,1)
(a) Find E[X]
(b) Find E[X²]
(c) Find E[X³]
(d) Using the formula Cov(X,Y) = E[XY] - E[X]E[Y] , show that Cov(X²,X) = 0
(e) Explain what this exercise tells us about the covariance and independence of two random variables.
Options:
A) {1}/{2}
B) {1}/{3}
C) 0
D) -{1}/{2}