Let the random vector X = (X₁, X2,..., Xm) have a multinomial joint probability mass function, in the sense: Pp (X = x) = Pp(X₁ = x₁, X₂ = x2,..., Xm = xm) with p = (p₁,...,Pm) and x = (a) Show that the joint k can be written as: (1,...,m) such that moment generating function: Mx (t) = Ep (etx) = Ep(eΣï=1¹i Xi) Mx (t) = Zi=1 Xi = n. m Σ i=1 Pieti n! II1*₂! n m II P j=1 (b) Notice that if Mx (t) is the joint moment generating function (MGF) for X then Mx (t₁,0,...,0) is the MGF for the marginal of X₁. Use now the result of point (a) for showing that marginal distribution of X₁, i = 1,...,n is the binomial distribution with parameters n and pi.
Let the random vector X = (X₁, X2,..., Xm) have a multinomial joint probability mass function, in the sense: Pp (X = x) = Pp(X₁ = x₁, X₂ = x2,..., Xm = xm) with p = (p₁,...,Pm) and x = (a) Show that the joint k can be written as: (1,...,m) such that moment generating function: Mx (t) = Ep (etx) = Ep(eΣï=1¹i Xi) Mx (t) = Zi=1 Xi = n. m Σ i=1 Pieti n! II1*₂! n m II P j=1 (b) Notice that if Mx (t) is the joint moment generating function (MGF) for X then Mx (t₁,0,...,0) is the MGF for the marginal of X₁. Use now the result of point (a) for showing that marginal distribution of X₁, i = 1,...,n is the binomial distribution with parameters n and pi.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please provide some explanation with the taken steps, thank you in advance. (the question is in the attached image)
Transcribed Image Text:Let the random vector X = (X₁, X2, ..., Xm) have a multinomial joint probability mass function,
in the sense:
with p
(a) 1
Pp(X = x) = Pp(X₁ = x1, X₂ = x2, ..., Xm = xm)
= (p₁,..., Pm) and x = (x₁,...,xm) such that 1 xi = n.
Show that the joint moment generating function:
can be written as:
=
m
Mx (t) = Ep (etx) = Ep (ei=1¹i Xi)
ti
m
Mx(t) = ( Speti
i=1
n
n!
m
m
¡II P
j=1
₁ xi!
xj
(b)
Notice that if Mx (t) is the joint moment generating function (MGF) for X then Mx (t₁,0,...,0)
is the MGF for the marginal of X₁. Use now the result of point (a) for showing that marginal
distribution of Xį, i = 1,...,n is the binomial distribution with parameters n and pi.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 6 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman