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Random Sums of Random Variables


Key Concepts

Key Concepts

  1. Let N be a random variable assuming positive integer values 1, 2, 3.... Let Xi be a sequence of independent random variables which are also independent of N and with E[Xi] = E[X] the same for all i. Then
     [N ] sum E Xi = E [N ]E[X] i=1
  2. Let N be a random variable assuming positive integer values 1, 2, 3.... Let Xi be a sequence of independent random variables which are also independent of N with Var[Xi] the same for i. Then
     [ ] sum N Var Xi = E [N ]Var [X] + (E [X])2Var [N ] i=1


Vocabulary

Vocabulary


Mathematical Ideas

Mathematical Ideas

This section is adapted from: S. K. Ross, Introduction to Probability Models, Third Edition, Academic Press, 1985, Chapter 3, pages 83-103.

Expectation of a Random Sum of Random Variables

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PG-13

Let N be a random variable assuming positive integer values 1, 2, 3.... Let Xi be a sequence of independent random variables which are also independent of N with common mean E[Xi] independent of i. (This is a weaker hypothesis than independent, identically distributed random variables which is the typical case in applications.) Then

 [N ] [ [N ]] sum sum E Xi = E E Xi |N = n i=1 [ [ i=1 ]] sum n = E E Xi |N = n i=1 sum oo [n sum ] = E Xi Pr [N = n] i=1 i=1 oo n sum sum = E [Xi]Pr [N = n] i= oo 1 i=n1 sum sum = E[X] Pr [N = n] i=1 i=1 sum oo = nE[X] Pr [N = n] i=1 ( sum oo ) = n Pr[N = n] E[X] i=1 = E [N ]E[X]

This is a special case of a more general theorem known as Wald’s equation.

Variance of a Random Sum of Random Variables

Let N be a random variable assuming positive integer values 1, 2, 3.... Let Xi be a sequence of independent random variables which are also independent of N with common mean E[X] and common variance Var[X] which doesn’t depend on i. (This is a weaker hypothesis than independent, identically distributed random variables which is the typical case in applications.) Then

 |_ _ | [N sum ] ( sum N )2 [N sum ] Var Xi = E |_ Xi _| - (E Xi )2 i=1 i=1 i=1

We know how to evaluate the second term. Let’s work on the first term. Condition on N.

 |_ ( N )2_ | |_ |_ ( N )2 _ | _| |_ sum _| |_ |_ sum _| _| E Xi = E E Xi |N = n i=1 i=1

Now work on the innermost conditional expectation and use that we are conditioning on N = n. Use the identity that E[Y 2] = Var[Y ] + (E[Y ])2, so (also using linearity of conditional expectations)

 |_ _ | |_ _| ( N sum )2 ( sum n )2 E |_ X |N = n _| = E |_ X _| i i i=1 [ i=1 ] [ ] sum n sum n = Var Xi + (E Xi )2 i=1 i=1 2 = n Var[Xi] + (nE[X])

Notice that in the last step, we used that the variance of a sum of independent random variables is the sum of the variances. Now perform the outer expectation with respect to N.

 |_ |_ ( N )2 _| _ | oo |_ |_ sum _| _| sum 2 E E Xi |N = n = (n Var[X] + (nE[X]) ) Pr[N = n] i=1 i=1 = E [N ]Var[X] + E [N 2]E[X]2

Hence

 [ ] |_ ( )2_ | ( [ ])2 N sum sum N sum N Var Xi = E |_ Xi _| - E Xi i=1 i=1 i=1 [ 2] 2 2 = E [N ]Var[X] + E N E[X] - (E [N ]E[X]) = E [N ]Var[X] + E[X]2 (E [N 2] - E [N ]2) 2 = E [N ]Var [X] + (E [X]) Var [N ]

Problems to Work for Understanding

  1. The number of customers entering a store on a given day is Poisson distributed with mean c = 10. The amount of money spent by a customer is uniformly distributed over (0, 100). Find the mean and variance of the amount of money that the store takes in on a given day.

    Solution

  2. The number of claims received at an insurance company each week is a random variable with mean m1 and variance s12. The amount paid on each claim is a random variable with mean m2 and variance s22. Find the mean and variance of the amount of money paid by the insurance company each week. What assumptions are you making? Give a justification for why these assumptions are reasonable, and provide a reasonable scenario where these assumptions may not be reasonable.


Reading Suggestion:

  1. S. K. Ross, Introduction to Probability Models, Third Edition, Academic Press, 1985, Chapter 3, pages 83-103.

Outside Readings and Links:


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