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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].

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter14: Counting And Probability
Section14.2: Probability
Problem 3E: The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the...
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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).
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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