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)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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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 \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01fc7e81-378c-449d-b29e-55849dfe9053%2F215e92a5-8ae2-4051-a589-3fe5cf0282d9%2Fu7o4ys_processed.png&w=3840&q=75)
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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