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 X4, i > 1. Let N Y =) Xị. 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 (i) and the results from part (ii) show that Var[Y] = E[N]Var[X] + (E[X])*Var[N].
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 X4, i > 1. Let N Y =) Xị. 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 (i) and the results from part (ii) show that Var[Y] = E[N]Var[X] + (E[X])*Var[N].
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![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,.
i=1
Q2(i.) Let øy (t) = E[etY] be the moment generating function of the random variable Y. Using conditioning
argument (i.e., E[X] = E[E[X[Y]) show that
Øy (t) = E[(¢x(t))^],
where ox (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 (i) and the results from part (ii) 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;
i=1
is exponentially distributed with parameter Ap using the moment generating function from part (i).
Hint: The m.g.f. of an exponential random variable with parameter A is X/(A – t).
Q(3.) (Continuation of Q2) 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)"¯), then show that
N
Y = ) X;
i=1
is exponentially distributed with parameter Xp using conditioning arguments directly with the random sum, instead
of using the m.g.f. method.
Hint: Write the C.D.F of the Gamma random variable in terms of the Poisson process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F5bb3522d-0ab3-47c4-b1e6-d3b4c208f5a4%2F51dgpjg_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,.
i=1
Q2(i.) Let øy (t) = E[etY] be the moment generating function of the random variable Y. Using conditioning
argument (i.e., E[X] = E[E[X[Y]) show that
Øy (t) = E[(¢x(t))^],
where ox (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 (i) and the results from part (ii) 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;
i=1
is exponentially distributed with parameter Ap using the moment generating function from part (i).
Hint: The m.g.f. of an exponential random variable with parameter A is X/(A – t).
Q(3.) (Continuation of Q2) 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)"¯), then show that
N
Y = ) X;
i=1
is exponentially distributed with parameter Xp using conditioning arguments directly with the random sum, instead
of using the m.g.f. method.
Hint: Write the C.D.F of the Gamma random variable in terms of the Poisson process.
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