**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) $:

Introduction: The joint density function of two random variables X and Y is given below:

Answered: Suppose that X and Y are continuous…

Math

Statistics

Suppose that X and Y are continuous random variables with joint pdf f (x, y) = e-**) 0

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Question

**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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