Let (X, Y ) be independent and uniformly distributed random variables (RVs) on the interval [0, 1]. Equivalently, let (X, Y ) be a point chosen uniformly at random in the unit box [0, 1] × [0, 1] on the (x, y) plane. Define the RV Z ≜ √X^2−Y^2 as the random distance of the point (X, Y ) from the origin (0, 0). Find the probability density function (PDF) for the RV Z.

Let (X, Y ) be independent and uniformly distributed random variables (RVs) on the interval [0, 1]. Equivalently, let (X, Y ) be a point chosen uniformly at random in the unit box [0, 1] × [0, 1] on the (x, y) plane. Define the RV Z ≜ √X^2−Y^2 as the random distance of the point (X, Y ) from the origin (0, 0). Find the probability density function (PDF) for the RV Z.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 30E

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Let (X, Y ) be independent and uniformly distributed random variables (RVs) on the interval[0, 1]. Equivalently, let (X, Y ) be a point chosen uniformly at random in the unit box [0, 1] × [0, 1] on the(x, y) plane. Define the RV Z ≜ √X^2−Y^2 as the random distance of the point (X, Y ) from the origin (0, 0). Find the cumulative distribution function (CDF) for the RV Z.
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