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c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw e Ω,Ζ(ω) -Χ(ω) + Υ(ω) . i) Show that: P(Z = z) = _fx,x(x,z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = fx(x)fv (z – x) = fx(z-y)fY (y) From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters x and Ay,respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay.

c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw e Ω,Ζ(ω) -Χ(ω) + Υ(ω) . i) Show that: P(Z = z) = _fx,x(x,z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = fx(x)fv (z – x) = fx(z-y)fY (y) From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters x and Ay,respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Part iii

c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw e Ω,Ζ(ω) = X(ω) + Y(ω) . i) Show that: P(Z = 2) = ) fx,x(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fr(z- x) = ) fx(z-y)fr(y) From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters Ax and Ay, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay.
Transcribed Image Text:c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw e Ω,Ζ(ω) = X(ω) + Y(ω) . i) Show that: P(Z = 2) = ) fx,x(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fr(z- x) = ) fx(z-y)fr(y) From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters Ax and Ay, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay.
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