1. ## Probability and Statistics

Hey I am stuck with two questions on my latest assignment and I'm really unsure of how to go about answering them!! I would be grateful for any help!!

1. Let P{X=a}=1 and P{Y=b}=1 for some numbers a and b. Prove that X and Y are independent.

2. Let X and Y be independent identically distributed random variables, EX=EY=mu, EX^2=EY^2=mu2, VarX=VarY=sigma^2,EX^3=EY^3=mu3
i) Show that E(X+Y-2mu)(X-Y)=0. Conclude that (X+Y) and (X-Y) are uncorrelated.
ii) Show that E(X-Y)^2=2sigma^2
iii) Show that E(X+Y)(X-Y)^2=2mu3-2(sigma^2)mu -2mu^3
iv) Conclude that (X+Y) and (X-Y) are not independent if mu3 is not equal to 3(sigma^2)mu +mu^3

Thank you!!

2. ## Re: Probability and Statistics

1) what's the probability that both a and b occur? does this equal the product of the individual probabilities?

2)
i) must multiply it out and take expectations. You're given all the values you'll need
ii) same as i
iii) same as ii
iv) if they aren't uncorrelated they cannot be independent (not the converse is not true)