Let X and Y have a joint pdf given by [3x, 0≤ y ≤x≤ 1, 10, otherwise. f(x, y) = = 1. Find the marginal pdf's of X and Y. 2. Find P(X ≤ 3/4 | Y ≤ 1/2). 3. Let T = X+Y. Find E[T].
Let X and Y have a joint pdf given by [3x, 0≤ y ≤x≤ 1, 10, otherwise. f(x, y) = = 1. Find the marginal pdf's of X and Y. 2. Find P(X ≤ 3/4 | Y ≤ 1/2). 3. Let T = X+Y. Find E[T].
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...
Related questions
Question
Plz complete solution otherwise skip.
![**Title: Joint Probability Density Function Analysis**
**Joint Probability Density Function**
Let \( X \) and \( Y \) have a joint probability density function (pdf) given by:
\[ f(x,y) = \begin{cases}
3x, & 0 \leq y \leq x \leq 1, \\
0, & \text{otherwise}.
\end{cases} \]
**Problems:**
1. **Find the Marginal PDFs of \( X \) and \( Y \):**
Determine the marginal probability density functions of \( X \) and \( Y \).
2. **Find \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \):**
Calculate the conditional probability \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \).
3. **Find \( E[T] \) where \( T = X + Y \):**
Let \( T = X + Y \). Find the expected value \( E[T] \).
---
**Explanation of Terms and Steps:**
1. **Marginal PDFs:**
- The marginal pdf of \( X \), \( f_X(x) \), is found by integrating the joint pdf \( f(x,y) \) over all possible values of \( y \).
\[ f_X(x) = \int_{-\infty}^{\infty} f(x,y) \, dy \]
- The marginal pdf of \( Y \), \( f_Y(y) \), is found by integrating the joint pdf \( f(x,y) \) over all possible values of \( x \).
\[ f_Y(y) = \int_{-\infty}^{\infty} f(x,y) \, dx \]
2. **Conditional Probability:**
- The conditional probability \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \) requires the joint pdf and the marginal pdf of \( Y \).
3. **Expected Value \( E[T] \):**
- The expected value \( E[T] \) is calculated by using the definition of expectation for random variables \( X \) and \( Y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbfb1871-07e2-4dce-8005-269462079c4a%2F704a6909-663b-4ba4-b774-af56ea55a09c%2Fwmb0jp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Joint Probability Density Function Analysis**
**Joint Probability Density Function**
Let \( X \) and \( Y \) have a joint probability density function (pdf) given by:
\[ f(x,y) = \begin{cases}
3x, & 0 \leq y \leq x \leq 1, \\
0, & \text{otherwise}.
\end{cases} \]
**Problems:**
1. **Find the Marginal PDFs of \( X \) and \( Y \):**
Determine the marginal probability density functions of \( X \) and \( Y \).
2. **Find \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \):**
Calculate the conditional probability \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \).
3. **Find \( E[T] \) where \( T = X + Y \):**
Let \( T = X + Y \). Find the expected value \( E[T] \).
---
**Explanation of Terms and Steps:**
1. **Marginal PDFs:**
- The marginal pdf of \( X \), \( f_X(x) \), is found by integrating the joint pdf \( f(x,y) \) over all possible values of \( y \).
\[ f_X(x) = \int_{-\infty}^{\infty} f(x,y) \, dy \]
- The marginal pdf of \( Y \), \( f_Y(y) \), is found by integrating the joint pdf \( f(x,y) \) over all possible values of \( x \).
\[ f_Y(y) = \int_{-\infty}^{\infty} f(x,y) \, dx \]
2. **Conditional Probability:**
- The conditional probability \( P\left( X \leq \frac{3}{4} \mid Y \leq \frac{1}{2} \right) \) requires the joint pdf and the marginal pdf of \( Y \).
3. **Expected Value \( E[T] \):**
- The expected value \( E[T] \) is calculated by using the definition of expectation for random variables \( X \) and \( Y \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 5 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON