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2.1.1. Let f(r1, x2) = 4.012, 0 < 1 < 1, 0 < x2 < 1, zero elsewhere, be the p of X1 and X2. Find P(0< X1 < ,< X2 < 1), P(X1= X2), P(X1 < X2), ar P(X1 < X2). Hint: Recall that P(X1 X2) would be the volume under the surface f(x1, r2) 4x1x2 and above the line segment 0 < x1 = x2 < 1 in the x1 2-plane.

2.1.1. Let f(r1, x2) = 4.012, 0 < 1 < 1, 0 < x2 < 1, zero elsewhere, be the p of X1 and X2. Find P(0< X1 < ,< X2 < 1), P(X1= X2), P(X1 < X2), ar P(X1 < X2). Hint: Recall that P(X1 X2) would be the volume under the surface f(x1, r2) 4x1x2 and above the line segment 0 < x1 = x2 < 1 in the x1 2-plane.

MATLAB: An Introduction with Applications
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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**Problem Set 2.1.1: Probability Density Function Analysis** Let \( f(x_1, x_2) = 4x_1x_2 \), where \( 0 < x_1 < 1 \), \( 0 < x_2 < 1 \), and zero elsewhere, be the probability density function (pdf) of \( X_1 \) and \( X_2 \). You are asked to find: 1. \( P(0 < X_1 < \frac{1}{2}, \frac{1}{4} < X_2 < 1) \) 2. \( P(X_1 = X_2) \) 3. \( P(X_1 < X_2) \) 4. \( P(X_1 \leq X_2) \) **Hint:** Recall that \( P(X_1 = X_2) \) would be the volume under the surface \( f(x_1, x_2) = 4x_1x_2 \) and above the line segment \( 0 < x_1 = x_2 < 1 \) in the \( x_1x_2 \)-plane. In approaching these problems, consider the constraints given and focus on integrating over the specified regions in the plane. The hint prompts us to conceptualize \( P(X_1 = X_2) \) as a line segment integral. To solve the problems, you'll need to set up appropriate double integrals over the defined regions in the \( x_1x_2 \)-plane and compute the volumes where needed.
Transcribed Image Text:**Problem Set 2.1.1: Probability Density Function Analysis** Let \( f(x_1, x_2) = 4x_1x_2 \), where \( 0 < x_1 < 1 \), \( 0 < x_2 < 1 \), and zero elsewhere, be the probability density function (pdf) of \( X_1 \) and \( X_2 \). You are asked to find: 1. \( P(0 < X_1 < \frac{1}{2}, \frac{1}{4} < X_2 < 1) \) 2. \( P(X_1 = X_2) \) 3. \( P(X_1 < X_2) \) 4. \( P(X_1 \leq X_2) \) **Hint:** Recall that \( P(X_1 = X_2) \) would be the volume under the surface \( f(x_1, x_2) = 4x_1x_2 \) and above the line segment \( 0 < x_1 = x_2 < 1 \) in the \( x_1x_2 \)-plane. In approaching these problems, consider the constraints given and focus on integrating over the specified regions in the plane. The hint prompts us to conceptualize \( P(X_1 = X_2) \) as a line segment integral. To solve the problems, you'll need to set up appropriate double integrals over the defined regions in the \( x_1x_2 \)-plane and compute the volumes where needed.
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