23. Geometric generating function and moments. Using the notation and results of Exercise 22: a) Find the generating function of the geometric (p) distribution on {0, 1, 2, . }.
23. Geometric generating function and moments. Using the notation and results of Exercise 22: a) Find the generating function of the geometric (p) distribution on {0, 1, 2, . }.
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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![22. Factorial moments and the probability generating function. The kth factorial
moment of X is fk
distributions of X with range {0, 1, ...} it is easier to compute the factorial moments
than the ordinary moments µk =
coefficients Sn,k. These Sn,k are known as Stirling numbers of the second kind.
E[(X)k] where (X)k
X (х — 1)..- (х — к+1). For many
E[X*]. Note that x"
= E Sn,k(x)k for some integer
a) Find Sn,k for 1 < n < 3 and 1 < k < n.
b) Find a formula for µn in terms of fk, 1 <k< n.
c) Assuming X has non-negative integer values, let P(X = i) = pi for i = 0, 1,....
Let G(2) = E, Piz', known as the probability generating function of X.
Assume G(r) < ∞ for some r > 1. Show by switching the order of summation
and differentiation k times, (which can be justified, but you need not show this)
that the kth derivative G(k) (2) of the function G(z) is G(k) (2) = E Pi(i)kz*-'
Deduce that fr = G(k) (1).
23. Geometric generating function and moments. Using the notation and results of
Exercise 22:
a) Find the generating function of the geometric (p) distribution on {0,1, 2, .}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b586aea-5863-4664-9cfc-03ab6927a8aa%2Fd06173b4-59b2-456d-bfd6-9d1822ad5b56%2Focldcyt_processed.png&w=3840&q=75)
Transcribed Image Text:22. Factorial moments and the probability generating function. The kth factorial
moment of X is fk
distributions of X with range {0, 1, ...} it is easier to compute the factorial moments
than the ordinary moments µk =
coefficients Sn,k. These Sn,k are known as Stirling numbers of the second kind.
E[(X)k] where (X)k
X (х — 1)..- (х — к+1). For many
E[X*]. Note that x"
= E Sn,k(x)k for some integer
a) Find Sn,k for 1 < n < 3 and 1 < k < n.
b) Find a formula for µn in terms of fk, 1 <k< n.
c) Assuming X has non-negative integer values, let P(X = i) = pi for i = 0, 1,....
Let G(2) = E, Piz', known as the probability generating function of X.
Assume G(r) < ∞ for some r > 1. Show by switching the order of summation
and differentiation k times, (which can be justified, but you need not show this)
that the kth derivative G(k) (2) of the function G(z) is G(k) (2) = E Pi(i)kz*-'
Deduce that fr = G(k) (1).
23. Geometric generating function and moments. Using the notation and results of
Exercise 22:
a) Find the generating function of the geometric (p) distribution on {0,1, 2, .}.
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