Consider the function f(x,y)=c(x² + y²), a joint probability density function over 0 2.5 and Y <1.5) e. Are X and Y f. What is the correlation between X and Y? g. What is the probability that Y<1.5 given X-0 d. independent? You must demonstrate this using the definition.
Consider the function f(x,y)=c(x² + y²), a joint probability density function over 0 2.5 and Y <1.5) e. Are X and Y f. What is the correlation between X and Y? g. What is the probability that Y<1.5 given X-0 d. independent? You must demonstrate this using the definition.
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 do parts d,e,f,g
Transcribed Image Text:**Joint Probability Density Function Problem**
Consider the function \( f_{XY}(x,y) = c(x^2 + y^2) \), a joint probability density function over \( 0 < x < 5 \) and \( 0 < y < 5 \).
1. **Determine the value of \( c \)**:
Find the value of \( c \) that makes the function \( f_{XY}(x,y) \) a valid probability density function.
2. **Marginal Probability Density Function of \( X \), \( f_{X}(x) \)**:
Find the marginal probability density function of \( X \).
3. **Marginal Probability Density Function of \( Y \), \( f_{Y}(y) \)**:
Find the marginal probability density function of \( Y \).
4. **Joint Probability \( P(X > 2.5 \text{ and } Y < 1.5) \)**:
Calculate the probability that \( X \) is greater than 2.5 and \( Y \) is less than 1.5.
5. **Independence of \( X \) and \( Y \)**:
Determine whether \( X \) and \( Y \) are independent. You must demonstrate this using the formal definition.
6. **Correlation Between \( X \) and \( Y \)**:
Find the correlation between \( X \) and \( Y \).
7. **Conditional Probability**:
What is the probability that \( Y < 1.5 \) given \( X = 0 \)?
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