Berkeley STAT 134 - Conditional Probabilities, Distributions, & Expectations Review

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Problem 1Problem 2Problem 3Problem 4Stat 134: Conditional Probabilities, Distributions, &Expectations ReviewAdam LucasMay 8th, 2019Problem 1Let X1∼ Geom (p1), X2∼ Geom (p2), X1⊥ X2, both on {1, 2, . . .}.Find:a. P(X1≤ X2);b. P(X1= x | X1≤ X2). Recognize X1|X1≤ X2as a named distribu-tion, and state the parameter(s).Problem 2Let Y ∼ Beta (r, s). Conditioned on Y = y, let X ∼ Geometric (y) on{0, 1, 2, . . .} For simplicity, assume r , s > 1.a. What is E(X | Y = y)?b. Find E(X).c. Find P(X = x), for x ∈ {0, 1, 2, . . .}.sta t 134: conditional probabilities, distributions, & expectations review 2Problem 3Suppose a proportion p of a population has a gene m that makesthem predisposed to migraines. Of these people, the number of mi-graines they experience in a year follows a Poisson process with rateµ per year, whereas the rest of the population experiences migrainesaccording to a Poisson process with rate λ.a. What is the probability that a randomly selected individual expe-riences no migraines in a given year?Hint: Condition on whether the indi-vidual has gene m.b. Let Ntdenote the number of migraines a randomly selected indi-vidual experiences in t years. Find E(Nt).c. Find Va r(Nt).Problem 4Let X, Y have joint density fX,Y(x, y) = 2λ2e−λ(x+y), 0 < x < y. It canbe shown that fX(x) = 2λe−2λx, x > 0. Find:a. The conditional density of Y, given X = x;b. E(Y|X = x).Prepared by Brian


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Berkeley STAT 134 - Conditional Probabilities, Distributions, & Expectations Review

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