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3. Let X and Y be two r.v.'s with joint p.d.f. given by: fx,y (x, y): = [cye¯¤,0 < y ≤ x <∞, 0, otherwise (a) Determine the constant c. (b) Determine the marginal p.d.f.'s fx and fy and specify the range of the arguments involved. (c) Determine the conditional p.d.f.'s fx|y(Aly) and fy|x (Ax), and specify the range of the arguments involved.

3. Let X and Y be two r.v.'s with joint p.d.f. given by: fx,y (x, y): = [cye¯¤,0 < y ≤ x <∞, 0, otherwise (a) Determine the constant c. (b) Determine the marginal p.d.f.'s fx and fy and specify the range of the arguments involved. (c) Determine the conditional p.d.f.'s fx|y(Aly) and fy|x (Ax), and specify the range of the arguments involved.

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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(d) Calculate the conditional probability \( P(X > 2 \log 2 | Y = \log 2) \), where always log stands for the natural logarithm. (e) Show that \( E(E(X|Y)) = E(X) \). (f) Calculate the m.g.f. \( M_{X,Y} \).
Transcribed Image Text:(d) Calculate the conditional probability \( P(X > 2 \log 2 | Y = \log 2) \), where always log stands for the natural logarithm. (e) Show that \( E(E(X|Y)) = E(X) \). (f) Calculate the m.g.f. \( M_{X,Y} \).
3. Let \( X \) and \( Y \) be two random variables with joint probability density function (p.d.f.) given by: \[ f_{X,Y}(x, y) = \begin{cases} cye^{-x}, & 0 < y \leq x < \infty, \\ 0, & \text{otherwise} \end{cases} \] (a) Determine the constant \( c \). (b) Determine the marginal p.d.f.'s \( f_X \) and \( f_Y \) and specify the range of the arguments involved. (c) Determine the conditional p.d.f.'s \( f_{X|Y}(\Delta|y) \) and \( f_{Y|X}(\Delta|x) \), and specify the range of the arguments involved.
Transcribed Image Text:3. Let \( X \) and \( Y \) be two random variables with joint probability density function (p.d.f.) given by: \[ f_{X,Y}(x, y) = \begin{cases} cye^{-x}, & 0 < y \leq x < \infty, \\ 0, & \text{otherwise} \end{cases} \] (a) Determine the constant \( c \). (b) Determine the marginal p.d.f.'s \( f_X \) and \( f_Y \) and specify the range of the arguments involved. (c) Determine the conditional p.d.f.'s \( f_{X|Y}(\Delta|y) \) and \( f_{Y|X}(\Delta|x) \), and specify the range of the arguments involved.
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