Sums of Random Variables 2.2 2003/03/13 2003/04/11 Nick Kingsbury ngk10@cam.ac.uk Liqun Wang liqun@rice.edu Nick Kingsbury ngk10@cam.ac.uk Marginal Probability Random Variables This module introduces sums of random variables. Consider the random variable Y formed as the sum of two independent random variables X 1 and X 2 : Y X1 X2 where X 1 has pdf f1 x1 and X 2 has pdf f2 x2 . We can write the joint pdf for y and x 1 by rewriting the conditional probability formula: f y x1 f | y x1 f1 x1 It is clear that the event 'Y takes the value y conditional upon X1 x1 ' is equivalent to X 2 taking a value y x1 (since X2 Y X1 ). Hence f | y x1 f2 y x1 Now f y may be obtained using the Marginal Probability formula (this equation from this discussion of probability density functions). Hence f y x1 f | y x1 f1 x1 x1 f2 y x1 f1 x1 f2 f1 This result may be extended to sums of three or more random variables by repeated application of the above arguments for each new variable in turn. Since convolution is a commutative operation, for n independent variables we get: f y fn f n - 1 … f2 f1 fn f n - 1 … f2 f1 An example of this effect occurs when multiple dice are thrown and the scores are added together. In the 2-dice example of the subfigures a,b,c of this figure in the discussion of probability distributions, we saw how the pmf approximated a triangular shape. This is just the convolution of two uniform 6-point pmfs for each of the two dice. Similarly if two variables with Gaussian pdfs are added together, we shall show in the discussion of the summation of two or more Gaussian random variables that this produces another Gaussian pdf whose variance is the sum of the two input variances.