A random point (X,Y,Z)(X,Y,Z) is chosen uniformly from within the sphere S={(x,y,z), where x^2+y^2+z^2≤1}}. Because the joint distribution is a continuous one, and because the volume of a sphere of radius r is (4π/3)r^3, the joint probability density function of (X,Y,Z)is just f_X,Y,Z(x,y,z)=(3/4π) everywhere within the unit sphere, and 0 outside of it. Given this, what is the joint probability density function for (X,Y)? Restated, what is f_X,Y(x,y)? Derive and present a general analytic function for f_X,Y(x,y), and verify your answer by evaluating f_X,Y(1/11,1/3).

A random point (X,Y,Z)(X,Y,Z) is chosen uniformly from within the sphere S={(x,y,z), where x^2+y^2+z^2≤1}}. Because the joint distribution is a continuous one, and because the volume of a sphere of radius r is (4π/3)r^3, the joint probability density function of (X,Y,Z)is just f_X,Y,Z(x,y,z)=(3/4π) everywhere within the unit sphere, and 0 outside of it. Given this, what is the joint probability density function for (X,Y)? Restated, what is f_X,Y(x,y)? Derive and present a general analytic function for f_X,Y(x,y), and verify your answer by evaluating f_X,Y(1/11,1/3).

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

Section13.1: Continuous Probability Models

Problem 25E

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