Joint distribution for paired coin flips

I flip a coin three times and let

• X denote the number of heads on the first flip and
• Y denote the number of heads on the last two flips.

1. Write down the joint distribution p(x,y) of X and Y as a table
2. Fill out the margins to obtain p_X(x) and p_Y(y).
3. Compute E(XY).
4. Show that X and Y are independent.

1. Table

2. Table with Margins

3. E(XY) = (0 x 1/8)+(0 x 1/4)+(0 x 1/8)+(0 x 1/8)+(1 x 1/4)+(2 x 1/8) = 0.5

4. Check that E(XY) == E(X) * E(Y)
   E(XY) = 0.5
   E(X) = (1/8 x 1/4)+(1/8 x 1/4)+(1/4 x 1/2)+(1/4 x 1/2)+(1/8 x 1/4)+(1/8 x 1/4) = 0.375
   E(Y) = (1/8 x 1/2)+(1/4 x 1/2)+(1/8 x 1/2)+(1/8 x 1/2)+(1/4 x 1/2)+(1/8 x 1/2) = 0.5
   E(X) * E(Y) = 0.1875
   E(XY) =/= E(X) * E(Y) <-- Im not quite sure what I did wrong?

I like your computation of E(XY) but not your computation of E(X) or E(Y) so much. When you compute E(X), for example, you should only have two terms in the sum corresponding to the possible X values and you should use just the marginal probabilities - i.e. the far left and far right column in your table.

1. Table

2. Table with Margins

3. E(XY) = (0)(0)(1/8) + (0)(1)(1/4) + (0)(2)(1/8) + (1)(0)(1/8) + (1)(1)(1/4) + (1)(2)(1/8) = 1/2

4. E(X) = (0)(1/2) + (1)(1/2) = 1/2
   E(Y) = (0)(1/4) + (1)(1/2) + 2(1/4) = 1

   (E(X) * E(Y) = 1/2) and (E(XY) = 1/2)
   Therefore:
   X and Y are independent