Question

A fair coin is flipped twice. Let X be the number of heads in the two tosses, and Y be the indicator r.v. for the tosses landing the same way. That is, Y is equal to 1 if the tosses land in the same way, and 0 otherwise. (a) Find the joint pmf of X and Y.

Answer 1

Explanation:

It can be clearly observed that
`X\in{0,1,2}` and
`Y\in {0,1}`

Now lets find the probabilities of all possible combinations

`P(X=0, Y=0)=P(Y=0|X=0)P(X=0)=0`

Since it is impossible that the coin ha flipped in different ways given that all the time the tail has fallen

`P(X=1,Y=0)=P(Y=0|X=1)P(X=1)=1*2*((1)/(2) )^2` =1/2

`P(X=2,Y=0)=P(Y=0|X=2)P(X=2)=1*2*((1)/(2) )^2` =1/2

and similarly,

`P(X=0,Y=1)=P(Y=1|X=0)P(X=0)=1*((1)/(2) )^2= 1/4`

`P(X=1,Y=1)=P(Y=1|X=1)P(X=1)=0`

`P(X=2,Y=1)=P(Y=1|X=2)P(X=2)=1*((1)/(2) )^2=1/4`