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Written Problem 4: The gamma function which relates to probability and statistics is defined as: I(x)= [ r*e*dt. You may take it without proof that this improper integral converges when x is positive. Now, prove that: а) Г() 31 b) T(x+1)=x[(x) (Hint: Write the improper integral as a limit and use IbP). c) Using a) and b), show in a few lines that T(5)=4!, and briefly explain how the same reasoning would show that I(n)=(n-1)! for any positive integer n.

Written Problem 4: The gamma function which relates to probability and statistics is defined as: I(x)= [ r*e*dt. You may take it without proof that this improper integral converges when x is positive. Now, prove that: а) Г() 31 b) T(x+1)=x[(x) (Hint: Write the improper integral as a limit and use IbP). c) Using a) and b), show in a few lines that T(5)=4!, and briefly explain how the same reasoning would show that I(n)=(n-1)! for any positive integer n.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Written Problem 4:** The *gamma function* which relates to probability and statistics is defined as: \[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} \, dt. \] You may take it without proof that this improper integral converges when \( x \) is positive. Now, prove that: a) \( \Gamma(1) = 1 \) b) \( \Gamma(x+1) = x\Gamma(x) \) (Hint: Write the improper integral as a limit and use IbP). c) Using a) and b), show in a few lines that \( \Gamma(5) = 4! \), and briefly explain how the same reasoning would show that \( \Gamma(n) = (n-1)! \) for any positive integer \( n \).
Transcribed Image Text:**Written Problem 4:** The *gamma function* which relates to probability and statistics is defined as: \[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} \, dt. \] You may take it without proof that this improper integral converges when \( x \) is positive. Now, prove that: a) \( \Gamma(1) = 1 \) b) \( \Gamma(x+1) = x\Gamma(x) \) (Hint: Write the improper integral as a limit and use IbP). c) Using a) and b), show in a few lines that \( \Gamma(5) = 4! \), and briefly explain how the same reasoning would show that \( \Gamma(n) = (n-1)! \) for any positive integer \( n \).
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