Suppose that X and Y are continuous random variables with joint pdf f (x, y) = e-**) 0
Suppose that X and Y are continuous random variables with joint pdf f (x, y) = e-**) 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
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![**Introduction to Joint Probability Density Functions**
In this section, we will explore the concept of joint probability density functions (pdf) for continuous random variables through a given problem statement.
### Problem Statement:
Suppose that \( X \) and \( Y \) are continuous random variables with joint pdf
\[ f(x, y) = e^{-(x+y)} \]
\[ 0 < x < \infty \text{ and } 0 < y < \infty \]
and zero otherwise.
Given this information, we aim to find the following probabilities:
a. Find \( \mathbb{P}(X > 3) \).
b. Find \( \mathbb{P}(X > Y) \).
c. Find \( \mathbb{P}(X + Y > 3) \).
### Explanation of the Joint PDF \[ f(x, y) \]:
The function \( f(x, y) = e^{-(x+y)} \) describes the joint probability density function of \( X \) and \( Y \). This indicates that the probability distribution is governed by the exponential of the negative sum of \( x \) and \( y \). The domain is restricted to positive \( x \) and \( y \), specifically \( 0 < x < \infty \) and \( 0 < y < \infty \). Outside this domain, the pdf is zero meaning no probability density is present outside the specified range.
**Visualizations:**
1. **Function Plot:** Imagine a 3D plot with \( X \) and \( Y \) on the horizontal axes and \( f(x, y) \) on the vertical axis. The plot will show that as either \( x \) or \( y \) increases, the value of \( f(x, y) \) decreases exponentially, indicating diminishing probability density further from the origin.
### Solution Approach:
To effectively solve the given queries, we will integrate the joint pdf within the specified boundaries, adhering to the constraints provided.
**a.** Finding \( \mathbb{P}(X > 3) \):
\[ \mathbb{P}(X > 3) = \int_{3}^{\infty} \left( \int_{0}^{\infty} e^{-(x+y)} \, dy \right) dx \]
**b.** Finding \( \mathbb{P}(X > Y) \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa512e5ed-0cb1-424e-ac94-5a6d2cf189ee%2F29f4392c-f536-4a1a-baca-55a8bd42c715%2Ftujp4iq.png&w=3840&q=75)
Transcribed Image Text:**Introduction to Joint Probability Density Functions**
In this section, we will explore the concept of joint probability density functions (pdf) for continuous random variables through a given problem statement.
### Problem Statement:
Suppose that \( X \) and \( Y \) are continuous random variables with joint pdf
\[ f(x, y) = e^{-(x+y)} \]
\[ 0 < x < \infty \text{ and } 0 < y < \infty \]
and zero otherwise.
Given this information, we aim to find the following probabilities:
a. Find \( \mathbb{P}(X > 3) \).
b. Find \( \mathbb{P}(X > Y) \).
c. Find \( \mathbb{P}(X + Y > 3) \).
### Explanation of the Joint PDF \[ f(x, y) \]:
The function \( f(x, y) = e^{-(x+y)} \) describes the joint probability density function of \( X \) and \( Y \). This indicates that the probability distribution is governed by the exponential of the negative sum of \( x \) and \( y \). The domain is restricted to positive \( x \) and \( y \), specifically \( 0 < x < \infty \) and \( 0 < y < \infty \). Outside this domain, the pdf is zero meaning no probability density is present outside the specified range.
**Visualizations:**
1. **Function Plot:** Imagine a 3D plot with \( X \) and \( Y \) on the horizontal axes and \( f(x, y) \) on the vertical axis. The plot will show that as either \( x \) or \( y \) increases, the value of \( f(x, y) \) decreases exponentially, indicating diminishing probability density further from the origin.
### Solution Approach:
To effectively solve the given queries, we will integrate the joint pdf within the specified boundaries, adhering to the constraints provided.
**a.** Finding \( \mathbb{P}(X > 3) \):
\[ \mathbb{P}(X > 3) = \int_{3}^{\infty} \left( \int_{0}^{\infty} e^{-(x+y)} \, dy \right) dx \]
**b.** Finding \( \mathbb{P}(X > Y) \):
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