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where Dx = X₁-X₂ and Dy = |Y₁ — Y2]. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for D, and D, is (4(1-x)(1 − y), 0, fDxD, (x, y) = {4(1 − x) b. Define Ryx = D/Dx. Show that the pdf of Ryx is 2 fryx (r) = 1 33 1 3r3' 2 3r² 0≤x≤ 1,0≤ y ≤ 1 otherwise 0 ≤r≤1 1≤r≤00

where Dx = X₁-X₂ and Dy = |Y₁ — Y2]. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for D, and D, is (4(1-x)(1 − y), 0, fDxD, (x, y) = {4(1 − x) b. Define Ryx = D/Dx. Show that the pdf of Ryx is 2 fryx (r) = 1 33 1 3r3' 2 3r² 0≤x≤ 1,0≤ y ≤ 1 otherwise 0 ≤r≤1 1≤r≤00

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 20E
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4. Consider a square service region of unit area in which travel is right angle and directions of travel are parallel to the sides of the square. Let (X, Y₁) be the location of a mobile unit and (X₂, Y₂) the location of a demand for service. The travel distance is D =Dx + Dy where Dx = |X₁ - X₂ and Dy = |Y₁ — Y2\. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for Dx and Dy is (4(1-x)(1-y), fpx.D¸y (x, y) = {4(1 – b. Define Ryx = D/Dr. Show that the pdf of Ryx is 2 3 fryx (r) = . 2 3r² 1 r, 3 1 3r3' 0, 0≤x≤ 1,0 ≤ y ≤ 1 otherwise 0 ≤r≤1 1 ≤r <∞0
Transcribed Image Text:4. Consider a square service region of unit area in which travel is right angle and directions of travel are parallel to the sides of the square. Let (X, Y₁) be the location of a mobile unit and (X₂, Y₂) the location of a demand for service. The travel distance is D =Dx + Dy where Dx = |X₁ - X₂ and Dy = |Y₁ — Y2\. Assume that the two locations are independent and uniformly distributed over the square. a. Show that the joint pdf for Dx and Dy is (4(1-x)(1-y), fpx.D¸y (x, y) = {4(1 – b. Define Ryx = D/Dr. Show that the pdf of Ryx is 2 3 fryx (r) = . 2 3r² 1 r, 3 1 3r3' 0, 0≤x≤ 1,0 ≤ y ≤ 1 otherwise 0 ≤r≤1 1 ≤r <∞0
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Step 1: Distribution of product and ratio of two random variables obtained.

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