Question

Let X be the number of heads and let Y be the number of tails in 6 flips of a fair coin. Show that E(X · Y ) 6= E(X)E(Y ).

Answer 1

We have, X is the number of heads and Y is the number tails in 6 flips of a fair coin. Since, the number of heads and the number of tails can be any integer between 0 and 6, so we have X = 0(1)6 and Y = 0(1)6.

So, both X and Y individually follow binomial distribution with parameters n = 6 and p = 1/2. The probability mass function of X and Y are given by,

and

Using this we prepare the joint probability distribution chart of X and Y

X=x	Y=y	P(X=x)	P(Y=y)	P(X=x,Y=y)	x.P(X=x)	y.P(Y=y)	x.y.P(X=x,Y=y)
0	6	1/64	1/64	1/64	0	3/32	0
1	5	3/32	3/32	3/32	3/32	15/32	15/32
2	4	15/64	15/64	15/64	15/32	15/16	15/8
3	3	5/16	5/16	5/16	15/16	15/16	45/16
4	2	15/64	15/64	15/64	15/16	15/32	15/8
5	1	3/32	3/32	3/32	15/32	3/32	15/32
6	0	1/64	1/64	1/64	3/32	0	0
Total		1	1	1	3	3	15/2

From the defintions of E(X.Y), E(X) and E(Y) we have,

= 15/2

and   

= 3

and

= 3

Thus we have

The question seems to be wrong.