Let be a positive integer random variable with PMF of the form

()=122,=1,2,....

Once we see the numerical value of , we then draw a random variable whose (conditional) PMF is uniform on the set {1,2,...,2}.

1.1

Write down an expression for the joint PMF ,(,).

For =1,2,... and =1,2,...,2:

,(,)=?

1.2

Find the marginal PMF () as a function of . For simplicity, provide the answeronly for the case whenis an even number. (The formula for when is odd would be slightly different, and you do not need to provide it).

Hint:You may find the following helpful: =0=11 for 0<<1.

For =2,4,6,...:

()=?

1.3.

Let be the event that is even. Find (|=) and ().

(=)= ?

()=?

1.4 Is the event independent of ?