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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].

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
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**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 \( Y \).
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 \( Y \).
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