Answer the following questions for the random variables X and Y that have a bivariatenormal distribution and whose joint density function is f(x, y) as shown below. The jointdensity function has not been completely simplified or presented as a piecewise functionfor this questionf(x, y)=-3.8580 (10)-1.20(1)(+0]e6.2470.5904a. What is the marginal density function for the random variable X? Leave your answeras a piecewise function.b. The random variable X has ac. The distribution for the random variable X has a mean ofstandard deviation ofvariables X and Y isdistribution.and the covariance for the randomwith a

Given: Two random variables X and Y are said to have a bivariate normal distribution. Here, the…

Answered: Answer the following questions for the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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.)
Let Y1 = 0.5, Y2 = 0.25, Y3 = 0.75, Y4 = 0.25 and Y5 = 1.25 be a random sample of width 5 selected from the population with the following probability density function. Which of the following is the estimation value obtained by the moment method for the unknown q parameter of this population?
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 Y1, . . . , Yn is a random sample from a population whose density function is
A civil engineer is studying the live loads on a highway bridge. Let Y and Z denote the live load (in ten thousands of pounds) for traffic going east and west, respectively, at a randomly selected time of day. Suppose these random variables have a joint distribution with density function f(y,z) = (1/400)yz2e–y/5–z/2, for y > 0, z > 0. What type of distributions do Y and Z have? Use this to confirm the mean and standard deviation of each variable.
Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?
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 the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [4.9, 5.1] mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?
On a production line, parts are produced with a certain average size, but the exact size of each part varies due to the imprecision of the production process. Suppose that the difference between the size of the pieces produced (in millimeters) and the average size, which we will call production error, can be modeled as a continuous random variable X with a probability density function given by f(x) = 2, 5e^(-5|x|), for x E R (is in the image). Parts where the production error is less than -0.46 mm or greater than 0.46 mm should be discarded. Calculate (approximating to 4 decimal places): a) What is the proportion of parts that the company discards in its production process? b) What is the proportion of parts produced where the production error is positive? c) Knowing that for a given part the production error is positive, what is the probability of this part being discarded?
Find the mean and variance for the distribution of random variable X whose density function is f(x). f(x,y)= 1/16x^2e^-x/2 x>0
Suppose that the random variable B has the standard normal density. What is the conditional probability density function of the sum of the two roots of the quadratic equation x2 + 2Bx + 1 = 0 given that the two roots are real? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTION
Let Y be a random variable whose density function is 3y2 ? y3? f(y|θ)= θ3 exp −θ3 , y>0,where θ > 0 is a parameter. Using an appropriate pivotal quantity, verify that?Y,Y? (− log α)1/3 (− log(1 − α))1/3is a confidence interval for θ with coverage probability 1 − 2α.

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