LECTURE 13ABookstore Example (1)Sum of a Random Number ofIndependent Random VariablesBookstore Example (2)Review of TransformsTransform of “Random Sum”Bookstore Example (3)Motivational ExampleBranching Process: MeanBranching Process: VarianceBranching Process: TransformsChallengeLECTURE 13A• Readings: Section 4.3, 4.4Lecture outline• Sum of a random number ofindependent random variables:– mean, variance, transformBookstore Example (1)• George visits a number of book stores looking for the “Hair Book”.• A bookstore caries such a book with probability .• The time George spends in each book store is exponentially distributed with .• George will visit bookstores until he finds the book.• We want to find the PDF, mean, variance of thetime he spends in bookstores.•Total time:Sum of a Random Number ofIndependent Random Variables• : nonnegative integer-valued r.v.• : i.i.d. r.v.s, independent of .• Let: . Then:• Mean:• Variance:Bookstore Example (2)• Number of bookstores, :– PMF– Mean– Variance(geometric, from n=1)• Time in each bookstore, (i.i.d., indep of ):– PDF– Mean– Variance•Total time, :– Mean– VarianceReview of Transforms• Definitions:• Moment generating properties:• Transform of sum of independent r.v.s:Transform of “Random Sum”• : nonnegative integer-valued r.v.• : i.i.d. r.v.s, independent of .• If , we have:• Compare with: • Thus, to get , start with and replace each occurrence of by .Bookstore Example (3)• Number of bookstores:– Transform(compare)• Time in each bookstore:– Transform•Total time:– Transform– PDF:(exponential, with λ = 1)Motivational Example• Branching Process:– Evolution, growth of a population of cells, increase of neutrons in a reactor, spread of an epidemic…• We need: mean, variance, PMF of .• and i.i.d. geometric, incl. zero.Branching Process: Mean• Recall:• For time step :• i.i.d.:• Mean (using previous slide):• Solve recursively, e.g.:Branching Process: VarianceBranching Process: Transforms• Recall, for time step :• Thus, to get , start with and replace each occurrence of by , where:Challenge•For•Show
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