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Z has the so-called one-sided Cauchy density (1+z²) 5.35 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/2. Verify that max (X, Y) has the same distribution as X + Y. 5.36 Suppose that X₁,..., Xn are independent random variables, each having

Z has the so-called one-sided Cauchy density (1+z²) 5.35 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/2. Verify that max (X, Y) has the same distribution as X + Y. 5.36 Suppose that X₁,..., Xn are independent random variables, each having

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 54E
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ve to wait? Suppose that X and Y are independent and uniformly distributed on (0, 1). What is the probability density of Z = X/Y? What is the probability that the first significant (nonzero) digit of Z equals 1? What about the digits 2.....9? 5.34 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/λ. Let Z = X/Y. Can you explain why the density function of Z does not depend on λ. Verify that Z has the so-called one-sided Cauchy density (1+2²) for z>0. 2 5.35 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/2. Verify that max(X, Y) has the same distribution as X + Y. 5.36 Suppose that X₁,..., Xn are independent random variables, each having uniform distribution on the interval (0, 1). (a) Use induction to prove that P(X₁ +...+X₁ ≤s) = (b) Let N be the smallest n such that X₁ + + X₂ > · ... for 0 ≤ s≤ 1. 1. Verify that E(N) = e. 5.37 A dealer draws successively random numbers from (0, 1) until the sum prespecified value a with 0 < a < 1. The dealer has to beat the hability that the dealer wins
Transcribed Image Text:ve to wait? Suppose that X and Y are independent and uniformly distributed on (0, 1). What is the probability density of Z = X/Y? What is the probability that the first significant (nonzero) digit of Z equals 1? What about the digits 2.....9? 5.34 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/λ. Let Z = X/Y. Can you explain why the density function of Z does not depend on λ. Verify that Z has the so-called one-sided Cauchy density (1+2²) for z>0. 2 5.35 Suppose that X and Y are independent random variables, each having an exponential distribution with expected value 1/2. Verify that max(X, Y) has the same distribution as X + Y. 5.36 Suppose that X₁,..., Xn are independent random variables, each having uniform distribution on the interval (0, 1). (a) Use induction to prove that P(X₁ +...+X₁ ≤s) = (b) Let N be the smallest n such that X₁ + + X₂ > · ... for 0 ≤ s≤ 1. 1. Verify that E(N) = e. 5.37 A dealer draws successively random numbers from (0, 1) until the sum prespecified value a with 0 < a < 1. The dealer has to beat the hability that the dealer wins
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