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2. Let the bivariate continuous random vector (X,Y) have the following joint probability density function: {; c 0

2. Let the bivariate continuous random vector (X,Y) have the following joint probability density function: {; c 0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Only need part(d), (e), (f), thank you for your help!

2. Let the bivariate continuous random vector \( (X, Y) \) have the following joint probability density function: \[ f(x, y) = \begin{cases} c & 0 \leq y \leq 1 - x^2 \quad \text{and} \quad -1 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases} \] a. Graph the support on the two-dimensional X-Y plane, and find the value of \( c \). Hint: Integrate with respect to \( y \) first, \( dy \, dx \).
Transcribed Image Text:2. Let the bivariate continuous random vector \( (X, Y) \) have the following joint probability density function: \[ f(x, y) = \begin{cases} c & 0 \leq y \leq 1 - x^2 \quad \text{and} \quad -1 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases} \] a. Graph the support on the two-dimensional X-Y plane, and find the value of \( c \). Hint: Integrate with respect to \( y \) first, \( dy \, dx \).
b. Find the marginal probability density function of X, \( f_x(x) \). c. Find the marginal probability density function of Y, \( f_y(y) \). d. Find the conditional probability density function of X given \( Y=y \), \( f_{x|y}(x|y) \). e. Are X and Y independent? Why or why not. f. Find \( P(Y \leq X + 1) \). Do not work out the entire integral, just set up the double integral that needs to be solved and put in the correct limits. Hint: Sketch out the region using the support from part a.
Transcribed Image Text:b. Find the marginal probability density function of X, \( f_x(x) \). c. Find the marginal probability density function of Y, \( f_y(y) \). d. Find the conditional probability density function of X given \( Y=y \), \( f_{x|y}(x|y) \). e. Are X and Y independent? Why or why not. f. Find \( P(Y \leq X + 1) \). Do not work out the entire integral, just set up the double integral that needs to be solved and put in the correct limits. Hint: Sketch out the region using the support from part a.
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