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Homework 12 Extra Credit - one for simplicity and the...

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550.420 Introduction to Probability Fall 2011 Homework 12: Extra Credit Only Due November 21, 2011 (Monday before Thanksgiving) In honor of the flu season, consider the following problem dealing with a simplistic model for the spread of an epidemic. Consider a population of N individuals, through which an epidemic passes until all individuals have become infected. Suppose that at time 0 one member of the population is infected and the remaining individuals are susceptible. The only changes possible are when a susceptible individual becomes infected. Once infected, an individual remains infected, and infectious, thereafter. From the time when an individual becomes infected, the time until he or she infects each susceptible individual is an exponentially distributed random variable (with parameter
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Unformatted text preview: one, for simplicity), and the infection times of different pairs of individuals are independent of each other. Let the duration of the epidemic be denoted by 1 2 1-+ + + = N U U U T where U i is the amount of time spent with exactly i individuals infected. 1. (3 points) What is the distribution of U i for each possible value of i ? Justify your answer. 2. (2 points) Calculate E[T]. 3. (3 points) Does the expected duration of the epidemic increase or decrease as a function of N? Hint: Find an asymptotic expression for the expected duration, after using the identity . 1 1 1 ) ( 1 -+ =-K N K N K N K...
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