Suppose that X and Y have the joint probability mass functionP(X= x, Y = y)equals O otherwise).Let Z==x+y21 >XY, find P(Z > 4).x = 1, 2, 3 and y = 1, 2 (and

Given that Joint probability mass function of X and Y is f(X = x , Y = y) = (x + y)/21 , x = 1,2,3…

Answered: Suppose that X and Y have the joint…

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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Suppose that p(x, y), the joint probability mass function of X and Y, is given by p(1, 1) = 0.5, p(1,2)= 0.1, p(2, 1) = 0.1, p(2, 2) = 0.3 Calculate the conditional probability mass function of X given that Y = 1.
Assume b, c are independent random variables with uniform distribution on the interval [-L, L), where L e R'. Find the probability that the solutions of the quadratic equation x? + bx + c= 0 are real. What happens when L → +0?
2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0
3. Suppose that X and Y are independent with X ~ N(0, 1) and Y~ N(0,4). Let U = X + Y and V = X - Y. a) Find the joint probability density function of (U, V), fu,v(u, v). b) Are U and V independent?
b) Suppose that X₁ and X₂ have the joint probability density function defined as f(x₁, x₂) = (x1x²,0x₁51, 0 ≤ x₂ ≤1 elsewhere Find: i) the value of w that makes f(x₁, x₂) a probability density function. ii) the joint cumulative distribution function for X₁ and X₂. iii) P (X₂ ≤ | X₂ ≤ ³).
b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
The joint probability mass function of X and Y, p(x,y) is given by p(1,1)= p(2,1)= p(3,1)= p(1,2)=p(2,2)=0 p(3,2)= 18 p(1,3)=0 p(2,3)= p(3,3)=1² a) b) c) Calculate E(X) and Var(X). Compute E(X | Y = 1) and Var(X|Y=1). Are the random variables X and Y independent?
Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. FInd (d) fZ|XY (z|x,y), (e) fX|YZ(x|y, z).
3. Let random variables X and Y be independent with joint density fe(x, y). Let Ix (0) and Iy (0) be the Fisher information of X and Y, respectively. Prove that the Fisher information I(x,y)(0) = Ix(0) + Iy(0).
b) Suppose that X₁ and X₂ have the joint probability density function defined as ƒ(x1, x₂) = {W*1*2* (Wx₁x₂, 0≤x₁ ≤ 1, 0 ≤ x₂ ≤1 elsewhere Find: i) the value of w that makes f(x₁, x₂) a probability density function. ii) the joint cumulative distribution function for X₂ and X₂. iii) P (X₁ ≤ 1X₂ ≤²).

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Suppose that X and Y have the joint probability mass function x+y = 1,2,3 and y = 1, 2 (and P(X= x, Y = y) = 21, equals O otherwise). Let Z = XY, find P(Z > 4).