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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,
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.
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?