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Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E Q, Z(@) = X(@) + Y(@). i) Show that: P(Z = z) = _fx,v(x, z – x) - Z ii) Now assume that X and Y are independent. Show that: P(Z = 2) = ) fx(x)fr(z-x) = ) fx(z-y)fv(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. iv) Show that the conditional distribution of X, given X + Y = n, is binomial, and find its parameters.

Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E Q, Z(@) = X(@) + Y(@). i) Show that: P(Z = z) = _fx,v(x, z – x) - Z ii) Now assume that X and Y are independent. Show that: P(Z = 2) = ) fx(x)fr(z-x) = ) fx(z-y)fv(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. iv) Show that the conditional distribution of X, given X + Y = n, is binomial, and find its parameters.

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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Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E Ω,Ζ() - X(ω) + Y(ω) . i) Show that: P(Z = z) = ) fx,y(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fy(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 ly, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + dy. iv) Show that the conditional distribution of X , given X + Y = n , is binomial, and find its parameters.
Transcribed Image Text:Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E Ω,Ζ() - X(ω) + Y(ω) . i) Show that: P(Z = z) = ) fx,y(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fy(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 ly, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + dy. iv) Show that the conditional distribution of X , given X + Y = n , is binomial, and find its parameters.
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