2. ( :) The random vector (X, Y) is said to be uniformly distributed over aregion R in the plane if, for some constant c, its joint density is1/5{Įc, if (x, y) ERf(x, y)0,otherwise.(a) Show that 1/c = area of region R.Suppose that (X,Y) is uniformly distributed over the square centered at(0,0), whose sides are of length 2.(b) Show that X and Y are independent, with each being distributed uniformlyover (-1, 1).(c) Whatat the origin? That is, find Prob(X² + Y² < 1).the probability that (X, Y) lies in the circle of radius 1 centered:::

Answered: Image /qna-images/answer/43e78d73-24b7-4b26-a3c7-b2e62e7992f1.jpg

Answered: 2. ( region R in the plane if, for some…

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...

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That is a lot of passwords! How many passwords would the website have if users were allowed to reuse the letters and numbers? (Enter the number of successful outcomes in each of the blanks below. Let the first six spaces represent letters and the last two spaces represent numbers) Answer • Answer • Answer • Answer • Answer • Answer • Answer • Answer = 30,891,577,600 That is almost Answer billion passwords!
Calculate the probability that X + Y < 3, where X and Y have joint prob- ability density (2xy +2x + y) 0 < x < 3, 0 < y < 3 p(x, y) = otherwise
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A system of random variables (X, Y) is normally distri- buted with the probability density ƒ(x, y) = 21/02 exp{-*² 2 +2 1². 20² Find the probability density of the system (R, D) if X = R cos , Y = R sin Þ.
7. Let u = (x, y) by a vector in R?. Draw the region in R² ((x, y)- plane) such that a) ||u||2 < 1, b) ||u||1 < 1, c) ||u|| < 1.
A sharpshooter is aiming at a circular target with radius 1. If we draw a rectangular system of coordinates with its origin at the center of the target, the coordinates of the point of impact, (X, Y), are random variables having the joint probability densityf(x,y) =⎧⎪⎪⎨⎪⎪⎩1/π for 0 < x^2 + y^2 < 10 elsewhereFind:(a) P[(X, Y) ∈ A], where A is the sector of the circle in the first quadrant bounded by the lines y = 0 and y = x;(b) P[(X, Y) ∈ B], where B = {(x, y)|0 < x2 + y2 < 1/2}
6. Consider a uniform rv U~ Unif[0, 1]. (a) Find the conditional distribution of U(U > a) for 0 < a < 1. (b) Find the conditional distribution of U(U < a) for 0 < a < 1. (c) If V~ Unif[0, 1] is independent of U, find the conditional density of V (U + V).
Let X, Y be random variables with joint density function f: a) Suppose Z = Y/X . Show that the density function of Z is (image 1) b) If X, Y are independent with distribution N(0, 1), then (image 2). Hint: Use part (a).
Ownership Age PE NoCat 13 Yes Cat 14 Yes NoCat 14 Yes Cat 19 Yes Cat 16 Yes Cat 13 Yes NoCat 12 Yes Cat 15 No Cat 18 Yes Cat 16 Yes Cat 17 Yes Cat 18 Yes Cat 16 No Cat 13 Yes NoCat 16 Yes Cat 12 Yes NoCat 11 No Cat 10 Yes Cat 13 Yes NoCat 14 No Cat 18 No NoCat 15 No NoCat 14 Yes Cat 16 Yes NoCat 12 No Cat 12 Yes Cat 17 Yes Cat 15 Yes Cat 14 Yes Cat 13 No Cat 17 Yes Cat 14 No Cat 14 Yes Cat 15 Yes Cat 17 No NoCat 18 Yes NoCat 13 Yes Cat 14 No Cat 14 Yes NoCat 15 Yes Cat 14 Yes Cat 16 Yes Cat 14 Yes NoCat 16 No NoCat 12 No NoCat 14 No NoCat 15 No NoCat 19 No NoCat 15 Yes NoCat 15 No Cat…

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