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 (b) U -V| (c) 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 (b) U -V| (c) min{U,V} (d) max{U,V}.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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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
(b) 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 () -n/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 = e\, and then sketch the graph of fy in the
special case that µ = 0 and o² = 1 (when X is standard Normal).
Note: Thie distribution is called Loa Normal(u s2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2F9aede6c4-7db1-49a5-a384-093246342850%2Fmcgd3f9_processed.jpeg&w=3840&q=75)
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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
(b) 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 () -n/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 = e\, and then sketch the graph of fy in the
special case that µ = 0 and o² = 1 (when X is standard Normal).
Note: Thie distribution is called Loa Normal(u s2).
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