A die is rolled and the number observed X is recorded Then a

A die is rolled and the number observed X is recorded. Then a coin is tossed number of times equal to the value of X. For example if X = 2 then the coin is tossed twice, etc. Let Y be the number of heads observed. Construct the joint probability distribution of X and Y. Find the conditional expected value of Y given X = 5. Find the conditional variance of Y given X = 5.

Solution

  E(Y/X=5)=1(5/32)+2(10/32)+3(10/32)+4(5/32)+5(1/32)+6(0)=80/32. V(Y/X=5)=[

1²(5/32)+2²(10/32)+3²(10/32)+4²(5/32)+5²(1/32)+6²(0)]-[1(5/32)+2(10/32)+3(10/32)+4(5/32)+5(1/32)+6(0)]²=240/32-(80/32)²=40.

X/Y	1	2	3	4	5	6	P(Y)
1	1/2	0	0	0	0	0	1/2
2	1/2	1/4	0	0	0	0	3/4
3	3/8	3/8	1/8	0	0	0	7/8
4	4/16	6/16	4/16	1/16	0	0	15/16
5	5/32	10/32	10/32	5/32	1/32	0	31/32
6	6/64	15/64	20/64	15/64	6/64	1/64	63/64
P(X)	120/64	94/64	64/64	29/64	8/64	1/64	321/64