(3) X and Y have the joint p.d.f: f(x,y) = { k, x + y ≤ 1; elsewhere. 0, (Hint: the region where f(x,y) #0 is a square with vertices (1,0), (0, 1), (−1,0), and (0, -1). Try to discuss |x|+|y| ≤ 1 in each quadrant and figure out why.) (a) Show that k = 1/2. (b) Find the marginal p.d.f. of X and marginal p.d.f. of Y. (c) Find the mean of X, μx and the mean of Y, µy. (d) Find the variance of X, o and the variance of Y, oz. (e) Find the covariance C[X,Y]. (f) Find the variance of X+Y, Ox+y. (g) Find the covariance C[X+Y, X-Y].
(3) X and Y have the joint p.d.f: f(x,y) = { k, x + y ≤ 1; elsewhere. 0, (Hint: the region where f(x,y) #0 is a square with vertices (1,0), (0, 1), (−1,0), and (0, -1). Try to discuss |x|+|y| ≤ 1 in each quadrant and figure out why.) (a) Show that k = 1/2. (b) Find the marginal p.d.f. of X and marginal p.d.f. of Y. (c) Find the mean of X, μx and the mean of Y, µy. (d) Find the variance of X, o and the variance of Y, oz. (e) Find the covariance C[X,Y]. (f) Find the variance of X+Y, Ox+y. (g) Find the covariance C[X+Y, X-Y].
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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Please this is a practice question. Please make sure to answer all parts. Thanks
![### Problem 3: Joint Probability Density Function of X and Y
X and Y have the joint p.d.f. given by:
\[
f(x, y) =
\begin{cases}
k, & |x| + |y| \leq 1; \\
0, & \text{elsewhere.}
\end{cases}
\]
**Hint:** The region where \( f(x, y) \neq 0 \) is a square with vertices \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\). Try to discuss \( |x| + |y| \leq 1 \) in each quadrant and figure out why.
### Tasks
(a) Show that \( k = \frac{1}{2} \).
(b) Find the marginal p.d.f. of X and the marginal p.d.f. of Y.
(c) Find the mean of X, \( \mu_X \), and the mean of Y, \( \mu_Y \).
(d) Find the variance of X, \( \sigma_X^2 \), and the variance of Y, \( \sigma_Y^2 \).
(e) Find the covariance \( C[X, Y] \).
(f) Find the variance of \( X + Y \), \( \sigma^2_{X+Y} \).
(g) Find the covariance \( C[X + Y, X - Y] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fed567308-85c0-4207-9c38-e7c148c2fdab%2F8dj3ljbf_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3: Joint Probability Density Function of X and Y
X and Y have the joint p.d.f. given by:
\[
f(x, y) =
\begin{cases}
k, & |x| + |y| \leq 1; \\
0, & \text{elsewhere.}
\end{cases}
\]
**Hint:** The region where \( f(x, y) \neq 0 \) is a square with vertices \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\). Try to discuss \( |x| + |y| \leq 1 \) in each quadrant and figure out why.
### Tasks
(a) Show that \( k = \frac{1}{2} \).
(b) Find the marginal p.d.f. of X and the marginal p.d.f. of Y.
(c) Find the mean of X, \( \mu_X \), and the mean of Y, \( \mu_Y \).
(d) Find the variance of X, \( \sigma_X^2 \), and the variance of Y, \( \sigma_Y^2 \).
(e) Find the covariance \( C[X, Y] \).
(f) Find the variance of \( X + Y \), \( \sigma^2_{X+Y} \).
(g) Find the covariance \( C[X + Y, X - Y] \).
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VIEWStep 5: Find the variance of X, and the variance of Y
VIEWStep 6: Find the covariance C[X,Y].
VIEWStep 7: Find the variance of X+Y, and find the covariance C[X + Y, X - Y]
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