6.15. The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is (c if(x, y) E R otherwise f(x, y) = a. Show that 1/c = area of region R. Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2. b. Show that X and Y are independent, with each being distributed uniformly over (-1,1). c. What is the probability that (X, Y) lies in the circle of radius 1 centered at the origin? That is, find P{X² + y² ≤ 1}.
6.15. The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is (c if(x, y) E R otherwise f(x, y) = a. Show that 1/c = area of region R. Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2. b. Show that X and Y are independent, with each being distributed uniformly over (-1,1). c. What is the probability that (X, Y) lies in the circle of radius 1 centered at the origin? That is, find P{X² + y² ≤ 1}.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![**6.15** The random vector \((X, Y)\) is said to be uniformly distributed over a region \(R\) in the plane if, for some constant \(c\), its joint density is
\[
f(x, y) =
\begin{cases}
c & \text{if } (x, y) \in R \\
0 & \text{otherwise}
\end{cases}
\]
a. Show that \(1/c = \text{area of region } R\).
Suppose that \((X, Y)\) is uniformly distributed over the square centered at \((0, 0)\) and with sides of length 2.
b. Show that \(X\) and \(Y\) are independent, with each being distributed uniformly over \((-1, 1)\).
c. What is the probability that \((X, Y)\) lies in the circle of radius 1 centered at the origin? That is, find \(P\{X^2 + Y^2 \leq 1\}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0618968-6b5a-406b-a92d-661b061825b7%2F172bfe07-9778-41af-83b3-4ce86ba2de08%2F5nz8bnk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**6.15** The random vector \((X, Y)\) is said to be uniformly distributed over a region \(R\) in the plane if, for some constant \(c\), its joint density is
\[
f(x, y) =
\begin{cases}
c & \text{if } (x, y) \in R \\
0 & \text{otherwise}
\end{cases}
\]
a. Show that \(1/c = \text{area of region } R\).
Suppose that \((X, Y)\) is uniformly distributed over the square centered at \((0, 0)\) and with sides of length 2.
b. Show that \(X\) and \(Y\) are independent, with each being distributed uniformly over \((-1, 1)\).
c. What is the probability that \((X, Y)\) lies in the circle of radius 1 centered at the origin? That is, find \(P\{X^2 + Y^2 \leq 1\}\).
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