Consider an image, in which every pixel takes a value of 1, with probability q, and a value 0, with probability 1q, where q is the realized value of a random variable Q which is distributed uniformly over the interval [0,1].

Let Xi be the value of pixel i. We observe, for each pixel the value of Yi=Xi+N, where N is normal with mean 2 and unit variance. (Note that we have the same noise at each pixel.) Assume that, conditional on Q, the Xi's are independent, and that the noise N is independent of Qand the Xi's.

Find E[Yi].

E[Yi]=

Find Var[Yi].

Var[Yi]=

Let A be the event that the actual values X1 and X2 of pixels 1 and 2, respectively, are zero. Find the conditional probability of Q given A.

For 0q1:

fQ|A(q)=