Answered: N Y =X; i=1 exponentially distributed… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 68E

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Q(3.) (Continuation of Q2) If X1, X2,-
... are independent and identically distributed exponential random variables
with parameter d 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 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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