2. The joint pdf of random variable x and y is 1 fxy(x, y) = 1,0 ≤ x ≤ y,0 ≤ y ≤2 2' Find (a) the marginal pdf f(x) and f(y), (b) the conditional pdfs f(xly) and fyx), and (c) E{xy=1} and E{xy=0.5}. (d) Are x and y independent? (e) find the correlation coefficient.
2. The joint pdf of random variable x and y is 1 fxy(x, y) = 1,0 ≤ x ≤ y,0 ≤ y ≤2 2' Find (a) the marginal pdf f(x) and f(y), (b) the conditional pdfs f(xly) and fyx), and (c) E{xy=1} and E{xy=0.5}. (d) Are x and y independent? (e) find the correlation coefficient.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem 2: Joint Probability Density Function**
The joint probability density function (pdf) of the random variables \( x \) and \( y \) is given by:
\[ f_{xy}(x, y) = \frac{1}{2}, \quad 0 \leq x \leq y, \, 0 \leq y \leq 2 \]
Tasks:
- (a) Find the marginal pdfs \( f_x(x) \) and \( f_y(y) \).
- (b) Find the conditional pdfs \( f(x|y) \) and \( f(y|x) \).
- (c) Calculate the expected values \( E\{x|y=1\} \) and \( E\{x|y=0.5\} \).
- (d) Determine if \( x \) and \( y \) are independent.
- (e) Find the correlation coefficient.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee255a41-3af6-4473-b85a-648a5fcbfc22%2F8dae5da9-e17d-453c-a096-3913668adb81%2Fq8wmun4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Joint Probability Density Function**
The joint probability density function (pdf) of the random variables \( x \) and \( y \) is given by:
\[ f_{xy}(x, y) = \frac{1}{2}, \quad 0 \leq x \leq y, \, 0 \leq y \leq 2 \]
Tasks:
- (a) Find the marginal pdfs \( f_x(x) \) and \( f_y(y) \).
- (b) Find the conditional pdfs \( f(x|y) \) and \( f(y|x) \).
- (c) Calculate the expected values \( E\{x|y=1\} \) and \( E\{x|y=0.5\} \).
- (d) Determine if \( x \) and \( y \) are independent.
- (e) Find the correlation coefficient.
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