Sum of Random Variables...

[Sitingbull]

Hello I have this following question and I am wondering if i am on the right path : here is the question

A picture in which pixel either takes 1 with a prob of q and 0 with a prob of 1-q, where q is the realized value of a r.v Q which is uniformly distributed in interval [0,1]

Let Xi be the value of pixel i, we observe for each pixel the value Yi = Xi + N where N is normal with mean 2 and unit variance.( Same noise distrib everywhere) Assume that conditional on Q the Xi's are independent and that the noise N is independent of Q and Xis, calculate the E(Yi) and the var(Yi).

For the expectation, i considered to be distributed unfi on [1,0] which means its expectation is 1/2 and and variance is 1/12.
So for the E(Yi) = it will be 1/2 + 2 = 2.5
for V(Yi) = 1/12 + 1 = 13/12

Is that correct or I am totally wrong on that?

Thank you in advance

 

[Valkarie]

Yes, your answer is correct. The expectation of Xi is 1/2 since it is uniformly distributed, and the variance is 1/12. When you add the noise N with mean 2 and variance 1, the expectation of Yi becomes 2.5, and the variance of Yi becomes 13/12.