Suppose that Y|X = x N(5 2x, 16), that is, given X = x, Y has the normal distribution with mean Y|X(x) =

Suppose that Y|X = x ∼ N(5 − 2x, 16), that is, given X = x, Y has the normal distribution with mean μY|X(x) = 5 − 2x and variance σ2

ε = 16, and let σX = 3.

(a) Let Y1 and Y2 be observations to be taken, independently from each other, when X has been observed to take the value 1 and 2, respectively. Find the probability that Y1 > Y2.

(b) Assume in addition that X has the normal distribution with mean 2 (and variance 9, as mentioned above). Use R commands to find the probability P(X ≤ 0,Y ≤ 2).