Let X and Y be independent random variables with the same geometric distribution. (a) Show that U and V are independent, where U and V are defined by U = min (X,Y) and V=X-Y. (b) Find the distribution of Z = X/(X+Y), where we define Z = 0 if X + Y = 0. (c) Find the joint pmf of X and X + Y.

Let X and Y be independent random variables with the same geometric distribution. (a) Show that U and V are independent, where U and V are defined by U = min (X,Y) and V=X-Y. (b) Find the distribution of Z = X/(X+Y), where we define Z = 0 if X + Y = 0. (c) Find the joint pmf of X and X + Y.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.2: Expected Value And Variance Of Continuous Random Variables

Problem 10E

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Question

Let X and Y be independent random variables with the same geometric distribution.
(a) Show that U and V are independent, where U and V are defined by
U = min (X,Y) and V = X-Y.
(b) Find the distribution of Z = X/(X+Y), where we define Z = 0 if X + Y = 0.
(c) Find the joint pmf of X and X + Y.

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