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A two-horse race is about to take place. Horse A's finishing time X is uniformly distributed between 3 and 8 minutes. Horse B's finishing time Y has a uniform distribution conditional on X, as follows: 4 x fx₁x (y|x) = for ≤ y ≤2x 7x 4 a. Find the pdf of Y (HINT: define first fxy and the joint sample of X and Y). b. Find fxy(xy), the conditional pdf for X given Y. C. Are the event S=X ≤ Y and the random variable Y independent, i.e. is P(S|Y=y) = P(S)? d. What is the probability that the winner wins by less than 1 minute? e. Find the probability that the winning time is less than 6 minutes.

A two-horse race is about to take place. Horse A's finishing time X is uniformly distributed between 3 and 8 minutes. Horse B's finishing time Y has a uniform distribution conditional on X, as follows: 4 x fx₁x (y|x) = for ≤ y ≤2x 7x 4 a. Find the pdf of Y (HINT: define first fxy and the joint sample of X and Y). b. Find fxy(xy), the conditional pdf for X given Y. C. Are the event S=X ≤ Y and the random variable Y independent, i.e. is P(S|Y=y) = P(S)? d. What is the probability that the winner wins by less than 1 minute? e. Find the probability that the winning time is less than 6 minutes.

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
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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1. A two-horse race is about to take place. Horse A's finishing time X is uniformly distributed between 3 and 8 minutes. Horse B's finishing time Y has a uniform distribution conditional on X, as follows: x fx₁x (y | x) = for = ≤ y ≤2x 4 7x 4 a. Find the pdf of Y (HINT: define first fxy and the joint sample of X and Y). b. Find fxy(xly), the conditional pdf for X given Y. c. Are the event S=X ≤ Y and the random variable Y independent, i.e. is P(S|Y=y) = P(S)? What is the probability that the winner wins by less than 1 minute? Find the probability that the winning time is less than 6 minutes. d. e.
Transcribed Image Text:1. A two-horse race is about to take place. Horse A's finishing time X is uniformly distributed between 3 and 8 minutes. Horse B's finishing time Y has a uniform distribution conditional on X, as follows: x fx₁x (y | x) = for = ≤ y ≤2x 4 7x 4 a. Find the pdf of Y (HINT: define first fxy and the joint sample of X and Y). b. Find fxy(xly), the conditional pdf for X given Y. c. Are the event S=X ≤ Y and the random variable Y independent, i.e. is P(S|Y=y) = P(S)? What is the probability that the winner wins by less than 1 minute? Find the probability that the winning time is less than 6 minutes. d. e.
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