Conditional Expectations: Calculating E(Y|X=x) and E(X|Y=y)

Question

X1 and X2 are independent and uniformly distributed on {1,2,...,n}. Let X be the minimum and Y the maximum of X1 and X2. Calculate:

(a) E(Y|X=x)

(b) E(X|Y=y)

I tried making distribution tables for these, but it seemed like it would take forever to figure out. Is there an easier way I am missing?

Answer 1

Hint:

Notice that for $x<y$ you have $$\mathbb P[X = x, Y = y] = \mathbb P[X_1 = x]\mathbb P[X_2 = y] + \mathbb P[X_2 = x]\mathbb P[X_1 = y] = \frac 2{n^2}$$ and for $x = y$ $$\mathbb P[X = x, Y = x] = \mathbb P[X_1 = x]\mathbb P[X_2 = x] = \frac 1{n^2}.$$

So you can calculate $\mathbb E(X|Y=y)$ and $\mathbb E(Y|X=y)$ from their definitions using the expressions above.