Homework Help is Here – Start Your Trial Now!
Math
Probability
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)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
See similar textbooks
icon
Related questions
Question
100%

How can I solve this question 6.15 ? 

**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 \(P\{X^2 + Y^2 \leq 1\}\).
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 \(P\{X^2 + Y^2 \leq 1\}\).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer