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Sums of Random Variables

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces sums of random variables.

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Consider the random variable YY formed as the sum of two independent random variables X1X1 and X2X2 :

Y= X1 + X2 Y X1 X2
(1)
where X1X1 has pdf f1 x1 f1 x1 and X2X2 has pdf f2 x2 f2 x2 .

We can write the joint pdf for yy and x1x1 by rewriting the conditional probability formula:

fy x1 =fy| x1 f1 x1 f y x1 f | y x1 f1 x1
(2)
It is clear that the event 'YY takes the value yy conditional upon X1 = x1 X1 x1 ' is equivalent to X2X2 taking a value y x1 y x1 (since X2 =Y X1 X2 Y X1 ). Hence
fy| x1 = f2 y x1 f | y x1 f2 y x1
(3)
Now fy f y may be obtained using the Marginal Probability formula ((Reference) from (Reference)). Hence
fy=fy| x1 f1 x1 d x1 = f2 y x1 f1 x1 d x1 = f2 * f1 f y x1 f | y x1 f1 x1 x1 f2 y x1 f1 x1 f2 f1
(4)
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:
fy= fn * f n - 1 ** f2 * f1 = fn * f n - 1 ** f2 * f1 f y fn f n - 1 f2 f1 fn f n - 1 f2 f1
(5)
An example of this effect occurs when multiple dice are thrown and the scores are added together. In the 2-dice example of a,b,c subfigure of (Reference) 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 (Reference) that this produces another Gaussian pdf whose variance is the sum of the two input variances.

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