2(i.) Let øy (t) = E[e*Y] be the moment generating function of the random variable Y. Using conditioning rgument (i.e., E[X] = E[E[X[Y]]) show that øy (t) = E[(øx(t))^], here øx(t) is the m.g.f. of X. 2(ii.) Using the moment generating function from part (i) show that E[Y] = E[N]E[X] 2(ii.) Using the moment generating function from part (i) and the results from part (ii) show that Var[Y] = E[N]Var[X] + (E[X])²Var[N].
2(i.) Let øy (t) = E[e*Y] be the moment generating function of the random variable Y. Using conditioning rgument (i.e., E[X] = E[E[X[Y]]) show that øy (t) = E[(øx(t))^], here øx(t) is the m.g.f. of X. 2(ii.) Using the moment generating function from part (i) show that E[Y] = E[N]E[X] 2(ii.) Using the moment generating function from part (i) and the results from part (ii) show that Var[Y] = E[N]Var[X] + (E[X])²Var[N].
MATLAB: An Introduction with Applications
6th Edition
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Author:Amos Gilat
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Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Could you solve (i), (ii), (iii)?
Thank you.
![Q2 Let X1, X2, -- be independent and identically distributed random variables. Let N be a non-negative, integer
valued random variable that is independent of the sequence X;, i > 1. Let
...
N
Y =X;.
Q2(i.) Let øy (t) = E[eY] be the moment generating function of the random variable Y. Using conditioning
argument (i.e., E[X] = E[E[X[Y]) show that
ør (t) = E[(øx(t))^],
where øx (t) is the m.g.f. of X.
Q2 (ii.) Using the moment generating function from part (i) show that
E[Y] = E[N]E[X]
Q2 (iii.) Using the moment generating function from part (1) and the results from part (i) show that
Var[Y]
= E[N]Var[X] + (E[X])²Var[N].
Q2(iv.) If X1, X2, ·. are independent and identically distributed exponential random variables with parameter A
and N is a geometric random variable with parameter p independent of the sequence X1, X2, ... (i.e
P(N = n) = p(1 – p)"-1), then show that
N
Y =X;
is exponentially distributed with parameter Xp using the moment generating function from part (i).
Hint: The m.g.f. of an exponential random variable with parameter A is A/(A – t).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2Fd265a4ba-51c3-4b74-acfc-a19fa55d781c%2Fblyiguj_processed.png&w=3840&q=75)
Transcribed Image Text:Q2 Let X1, X2, -- be independent and identically distributed random variables. Let N be a non-negative, integer
valued random variable that is independent of the sequence X;, i > 1. Let
...
N
Y =X;.
Q2(i.) Let øy (t) = E[eY] be the moment generating function of the random variable Y. Using conditioning
argument (i.e., E[X] = E[E[X[Y]) show that
ør (t) = E[(øx(t))^],
where øx (t) is the m.g.f. of X.
Q2 (ii.) Using the moment generating function from part (i) show that
E[Y] = E[N]E[X]
Q2 (iii.) Using the moment generating function from part (1) and the results from part (i) show that
Var[Y]
= E[N]Var[X] + (E[X])²Var[N].
Q2(iv.) If X1, X2, ·. are independent and identically distributed exponential random variables with parameter A
and N is a geometric random variable with parameter p independent of the sequence X1, X2, ... (i.e
P(N = n) = p(1 – p)"-1), then show that
N
Y =X;
is exponentially distributed with parameter Xp using the moment generating function from part (i).
Hint: The m.g.f. of an exponential random variable with parameter A is A/(A – t).
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