Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise Find the constant c, Find the marginal distribution functions(cdfs) for X and Y, Find the marginal density functions(pdfs) for X and Y, P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),

Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise Find the constant c, Find the marginal distribution functions(cdfs) for X and Y, Find the marginal density functions(pdfs) for X and Y, P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),

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 the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
Suppose that two continuous random variables X and Y have joint probability density function fxy = A( ex+y + e2x+y) , 1 ≤ x ≤ 2 ,0≤ y≤3 0 elsewhere a. P ( 3/2 ≤ X ≤ 2, 1 ≤ Y ≤ 2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0
For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?
Suppose X and Y are independent and identically distributed (i.i.d.) randomvariables, each with the uniform distribution on [0, 1]. What is the cumulative distributionfunction and the density function of XY ?
Suppose that two continuous random variables X and Y have a joint probability densityfunction f(x, y) = A(x − 3)y for -2≤x≤3 and 4≤y≤6a) What is the value of A?b) What is P(0≤x≤1 and 4≤y≤5)?c) Construct the marginal probability density functions.d) Are the random variables X and Y independent?e) If Y = 5, what is the conditional probability density function of X?f) What are the expectations and variances of the random variables X and Y ?g) What is the covariance of X and Y?h) What is the correlation between X and Y?
Let X and Y be a pair of continuous random variables with a joint density fx,y(x,y). Assume that fx,y(x,y) = cxy for x greater than or equal to 0, y greater than or equal to 0, and x + y less than or equal to 1. Here c is a constant. Assume that fx,y(x,y) is 0 elsewhere. What is the constant c equal to? With the value of c, what is E[XY]?
Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.
Let X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?
Determine the conditional probability distribution of Y given that X = 1. Where the jointprobability density function is given by f(x,y)=1/64xy for 0 < x < 4 and 0 < y < 4.
Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?
Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independent