Question

Let X be the number of heads in three tosses of a fair coin.

a. Find the probability distribution of Y = |X − 1|
b. Find the Expected Value of Y

Answer 1

In the toss of 3 coins, the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH and TTT. No. of heads obtained as below:

X: 0 1 2 3

P(X) 1/8 3/8 3/8 1/8

Thus X-1: -1 0 1 2

|X-1|: 1 0 1 2

P(|X-1|): 1/8 3/8 3/8 1/8

Here the probability of P(|X1|) will remain the same as corresponding to the above value.

Hence probability distribution of Y=|X-1|

P(Y=y) = 3/8 if y=0

=4/8 if y=1

= 1/8 if y= 2

b) Expected value of Y is

0*3/8 + 1*4/8 + 2* 1/8

=0+4/8 + 2/8 = 1/2 + 1/4 = 0.50 + 0.25 =0.75