(2) The joint p.d.f of the random variables X and Y is e-x-y, x>0, y > 0, { f(x, y)= 0, otherwise. (a) Find the marginal p.d.f.s of X and Y. (b) Find the mean of X, μx and the mean of Y, µy. (c) Find the variance of X, o and the variance of Y, oz. (d) Given Y = y, find the conditional p.d.f. of X, f(x|y). (e) Find E[X|y], for 0 < y. (f) Find o, for 0
(2) The joint p.d.f of the random variables X and Y is e-x-y, x>0, y > 0, { f(x, y)= 0, otherwise. (a) Find the marginal p.d.f.s of X and Y. (b) Find the mean of X, μx and the mean of Y, µy. (c) Find the variance of X, o and the variance of Y, oz. (d) Given Y = y, find the conditional p.d.f. of X, f(x|y). (e) Find E[X|y], for 0 < y. (f) Find o, for 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
please make sure to answer questions b,e,f
![### Problem Statement
**(2)** The joint probability density function (p.d.f) of the random variables \(X\) and \(Y\) is given by:
\[
f(x, y) =
\begin{cases}
e^{-x - y}, & x > 0, y > 0, \\
0, & \text{otherwise}.
\end{cases}
\]
**Tasks:**
(a) Find the marginal p.d.f.s of \(X\) and \(Y\).
(b) Find the mean of \(X\), \(\mu_X\), and the mean of \(Y\), \(\mu_Y\).
(c) Find the variance of \(X\), \(\sigma_X^2\), and the variance of \(Y\), \(\sigma_Y^2\).
(d) Given \(Y = y\), find the conditional p.d.f. of \(X\), \(f(x|y)\).
(e) Find \(E[X|y]\), for \(0 < y\).
(f) Find \(\sigma_X^2|y\), for \(0 < y\).
(g) Find the covariance \(C[X, Y]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F303b28e5-c90a-4f1a-b095-712878f553cb%2Fr3cfi6_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**(2)** The joint probability density function (p.d.f) of the random variables \(X\) and \(Y\) is given by:
\[
f(x, y) =
\begin{cases}
e^{-x - y}, & x > 0, y > 0, \\
0, & \text{otherwise}.
\end{cases}
\]
**Tasks:**
(a) Find the marginal p.d.f.s of \(X\) and \(Y\).
(b) Find the mean of \(X\), \(\mu_X\), and the mean of \(Y\), \(\mu_Y\).
(c) Find the variance of \(X\), \(\sigma_X^2\), and the variance of \(Y\), \(\sigma_Y^2\).
(d) Given \(Y = y\), find the conditional p.d.f. of \(X\), \(f(x|y)\).
(e) Find \(E[X|y]\), for \(0 < y\).
(f) Find \(\sigma_X^2|y\), for \(0 < y\).
(g) Find the covariance \(C[X, Y]\).
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