5. Let U,V be independent Uniform [0, 1] random variables. Find the CDF and PDF of each of the following random variables: (a) U V (6 \U – V| ©) min{U,V} d) max{U,V}.
5. Let U,V be independent Uniform [0, 1] random variables. Find the CDF and PDF of each of the following random variables: (a) U V (6 \U – V| ©) min{U,V} d) max{U,V}.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Question
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153%
5. Let U, V be independent Uniform [0, 1] random variables. Find the CDF and
PDF of each of the following random variables:
(a) U+V
U-V|
c) min{U,V}
(d) max{U,V}.
6. Using Poisson approximation, estimate how large a class needs to be so that
with probability > 99.9% some pair of students have the same birthday.
Note: Assume for simplicity that each student is equally likely to have been
born on any one of the 365 days in a year (ignoring leap years).
Hint: If there are n students, then there are () n²/2 pairs of students. Pairs
of students are not independent, however, they are approximately so. Use
Poisson approximation to estimate the probability that no pair of students
share the same birthday.
7. Suppose that X~
Normal (u, o?).
(a) Find the PDF fy (y) of Y = ex, and then sketch the graph of fy in the
special case that u = 0 and o? = 1 (when X is standard Normal).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2F091671f2-d8eb-4963-bc39-bf3831671fa0%2F7v635di_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3 / 3 |
153%
5. Let U, V be independent Uniform [0, 1] random variables. Find the CDF and
PDF of each of the following random variables:
(a) U+V
U-V|
c) min{U,V}
(d) max{U,V}.
6. Using Poisson approximation, estimate how large a class needs to be so that
with probability > 99.9% some pair of students have the same birthday.
Note: Assume for simplicity that each student is equally likely to have been
born on any one of the 365 days in a year (ignoring leap years).
Hint: If there are n students, then there are () n²/2 pairs of students. Pairs
of students are not independent, however, they are approximately so. Use
Poisson approximation to estimate the probability that no pair of students
share the same birthday.
7. Suppose that X~
Normal (u, o?).
(a) Find the PDF fy (y) of Y = ex, and then sketch the graph of fy in the
special case that u = 0 and o? = 1 (when X is standard Normal).
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