Stats exam questions I need help with

A discrete random variable takes on the values 0,1,2 and 3 with probabilities 1/8,1/4,1/8 and 1/2 respectively. Determine the pgf of the sum of two such random variables which are independent, stating any properties of the pgf you use.

The same type q comes up each year and I'm still unsure of how to tackle it? Any help would be much appreciated, test is on sat,

Re: Stats exam questions I need help with

Note that the possible sums of two numbers, each 0 to 3, are 0 to 6 and analyse the possible ways to get each number:

To get 0, both numbers would have to be 0: probability (1/8)(1/8)= 1/64.

To get 1, one number would have to be 0 and the other 1: The probability that the first is 0 and the second 1 is (1/8)(1/4)= 1/32 but so is the other order. The probability one number is 0 and the other 1 is 2(1/32)= 1/16.

To get 2, one number could be 0 and the other 2 or both could be 1. There are three ways to do that: (0, 2), (1, 1), and (2, 0). The probabilites of each are (1/8)(1/8)= 1/64, (1/4)(1/4)= 1/16, and (1/8)(1/8)= 1/64. The probability of a 2 is 1/64+1/16+ 1/64= 3/32.

To get 3, one number must be 3 and the other 0 or one number 2 and the other 1. There are four ways to do that: (0, 3), (3, 0), (2, 1), and (1, 2). Calculate the probabilities of each of those and add.

Continue for 4, 5, and 6. As a check make sure the sum of all six probabilities is 1.

Re: Stats exam questions I need help with

Thanks HallsofIvy!

Can I quiz you on another,

Let X denote the random variable that is the face value of a single die, ie P( X = k ) = 1/6. Let Y = 2X. Whatis the p.g.f of Y?

This seems almost too simple i think ive missed something? I know Y can have values 2,4,6,8,10,12. As its a uniform distribution wouldnt the pgf be P( Y= k ) = 1/6 as well?