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h. By completing the square, re-express fx,y (x, y) in the form fx,y(x, y) = e-(ax-by)² e-cy², Where a, b, c are constants. Note the difference between this expression and the original expression is that the second factor in this expression is a function of y and not of x. i. Find fy (y) using a technique similar to that you used in a.

h. By completing the square, re-express fx,y (x, y) in the form fx,y(x, y) = e-(ax-by)² e-cy², Where a, b, c are constants. Note the difference between this expression and the original expression is that the second factor in this expression is a function of y and not of x. i. Find fy (y) using a technique similar to that you used in a.

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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J) use your answer in i to find E[Y] and Var[Y] 
1. Suppose \( X \) and \( Y \) are jointly distributed with pdf \( f_{X,Y}(x,y) \), where \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{-(x-y)^2} e^{-x^2}. \] Note that \[ f_X(x) = \int_{-\infty}^{\infty} \frac{1}{\pi} e^{-(x-y)^2} e^{-x^2} \, dy. \]
Transcribed Image Text:1. Suppose \( X \) and \( Y \) are jointly distributed with pdf \( f_{X,Y}(x,y) \), where \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{-(x-y)^2} e^{-x^2}. \] Note that \[ f_X(x) = \int_{-\infty}^{\infty} \frac{1}{\pi} e^{-(x-y)^2} e^{-x^2} \, dy. \]
The text provides a mathematical problem involving expressions related to probability functions. Here is the transcription suitable for an educational website: --- **Problem h:** By completing the square, re-express \( f_{X,Y}(x,y) \) in the form \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{-(ax-by)^2} e^{-cy^2}, \] where \( a, b, c \) are constants. Note the difference between this expression and the original expression is that the second factor in this expression is a function of \( y \) and not of \( x \). **Problem i:** Find \( f_Y(y) \) using a technique similar to that you used in problem a. --- This section is guiding students to manipulate the joint probability distribution function \( f_{X,Y}(x,y) \) by completing the square. It emphasizes understanding the transformation of variables and highlights the key difference in the expression's dependency on \( y \) rather than \( x \). The subsequent problem asks students to find the marginal distribution \( f_Y(y) \) using a similar method.
Transcribed Image Text:The text provides a mathematical problem involving expressions related to probability functions. Here is the transcription suitable for an educational website: --- **Problem h:** By completing the square, re-express \( f_{X,Y}(x,y) \) in the form \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{-(ax-by)^2} e^{-cy^2}, \] where \( a, b, c \) are constants. Note the difference between this expression and the original expression is that the second factor in this expression is a function of \( y \) and not of \( x \). **Problem i:** Find \( f_Y(y) \) using a technique similar to that you used in problem a. --- This section is guiding students to manipulate the joint probability distribution function \( f_{X,Y}(x,y) \) by completing the square. It emphasizes understanding the transformation of variables and highlights the key difference in the expression's dependency on \( y \) rather than \( x \). The subsequent problem asks students to find the marginal distribution \( f_Y(y) \) using a similar method.
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