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Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0

Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 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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Suppose that \( X \) and \( Y \) are continuous random variables with joint pdf given by \[ f(x) = \begin{cases} c(x^2 + y^2) & 0 < x < 1, \, 0 < y < 1 \\ 0 & \text{elsewhere} \end{cases} \] 1. Find \( c \). 2. Find \( f_Y(y) \), the marginal density of \( Y \). 3. Find \( E(Y) \). 4. Find the conditional density of \( X \) given \( Y = y \). 5. Find \( P(X \leq \frac{1}{2} \mid Y = 1) \). 6. Are \( X \) and \( Y \) independent? Why or why not?
Transcribed Image Text:Suppose that \( X \) and \( Y \) are continuous random variables with joint pdf given by \[ f(x) = \begin{cases} c(x^2 + y^2) & 0 < x < 1, \, 0 < y < 1 \\ 0 & \text{elsewhere} \end{cases} \] 1. Find \( c \). 2. Find \( f_Y(y) \), the marginal density of \( Y \). 3. Find \( E(Y) \). 4. Find the conditional density of \( X \) given \( Y = y \). 5. Find \( P(X \leq \frac{1}{2} \mid Y = 1) \). 6. Are \( X \) and \( Y \) independent? Why or why not?
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