2. (a) When two random variables (X, Y) follow a bivariate normal distribution, thecovariance matrix Σ is defined asΣρσισ2(010₂ Posts),-=where p is the correlation, o, o are the variances of X and Y respectively. Use mvrnormfrom the MASS library (use library(MASS)) to generate 100 samples from a bivariate normalCall the first column xdistribution with μ = (₁, ₂) with μ₁ =(11).2,42= 3, and Σ ==and the second column y.(b) Plot the histogram of x and overlay it with the corresponding marginal normal density.Repeat for y. (Recall the marginal distribution of X is N (μ₁, 0²).)(c) Produce a scatter plot of x and y (use plot). Compute the sample correlation coefficient (usecor) and compare with the population correlation p. (First work out p in the Σ given.)

Answered: Image /qna-images/answer/f7af49b1-40fb-429e-b2ef-42db4146cc03.jpg

Answered: 2. (a) When two random variables (X, Y)…

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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Let wages denote hourly wages, educ years of education, and exper years of experience, and suppose log(wages) = f1 + 62educ + B3exper +e where E[eleduc, exper] = 0. Let b1, b2, and bz be our estimates for B1, B2, and ß3 and r23 denote the sample correlation between {educ,}", and {exper, }". Which of the following statements is true? II O a. The variance of b3 does not depend on r23. O b. The variance of b, is increasing in E" (educ, – educ,)² (all else equal). O c. The variance of b2 is at least as large as o²/ E", (educ, - educ,)2 for o? Var(eleduc, exper). O d. The variance of b2 depends on the sign of r23. O e. The variance of b2 always increases as r23 increases (all else equal).
9. (8') Let X,, X,, X, be a sample of size n=3 from a distribution with unknown mean u, -0 < u<0, where the variance o is a known positive number. Show that both à =i and ê, = 2X, + X, +5X, are unbiased estimators for H.
There are two random variables X and Y, and their correlation coefficient pX,Y = 0.7. Now, we have two new random variables A = 2.5X+1 and B = 4Y+2. Please compute the correlation coefficient of A and B, PA,B Please round your answer to one decimal place.
The quality characteristic of a product Y consists of three components X1, X2, and X3. The distributions of these components are normal with means 100, 2, 1 and variances 1, 1, 1 respectively. The specifications of the product are 90 ± 8. If Y= X1 + -7X2 + -9X3. Products that are below the lower specification limit are scrapped. Find the proportion of scrapped products = Ф(z).
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Let x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.
Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
Let S1^2 be the sampling variance for a random sample of twelve values (amount of mercury in the blood) and let S2^2 be the sampling variance for a random sample of ten values (amount of lead in the blood); samples from the same population were used. The population variance for mercury measurements is assumed to be twice the corresponding population variance for lead measurements. We will further assume that S1^2 is independent of S2^2. 1. Find a number b such that P [(S1^2/S2^2)<=b]=0.95 enter such a b to three decimal places b= 2. Consider the number b calculated above, find a number a such that P[a<=(S1^2/S2^2)<=b]=0.90 enter said a to three decimal places. a=
Let Y be a random variable with the mgfM(t) = .35e^(−4t) + .1 + .25e^(2t) + .3e^(4t).(a) What is the pmf of Y?(b) Calculate the variance of Y directly from the pmf you determined in part (a).(c) Calculate E[Y^4] directly from the mgf.
The sales manager of a large company selected a random sample of n = 10 salespeople and determined for each one the values of x = years of sales experience and y = annual sales (in thousands of dollars). A scatterplot of the resulting (x, y)pairs showed a linear pattern.(a)Suppose that the sample correlation coefficient is r = 0.75 and that the average annual sales is y = 100.If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales?Since the value for experience is 2 standard deviations above the mean experience, you can predict annual sales will be ________________standard deviations above the mean annual sales.(b)If a particular person whose sales experience is 1.7 standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of r? (Round your answer to two decimal places.)r =
A proton emitter produces proton beams with changing kinetic energy that is uniformly distributed between three and eight joules. Suppose that it is possible to adjust the upper limit of the kinetic energy (currently set to eight joules). • What is the mean kinetic energy? What is the variance of the kinetic energy? What is the probability that a proton beam has a kinetic energy of exactly 3.5 joules? a. E(X) = 5.5, V(X) = 2.0833, probability is zero (0) because X has a continuous distribution O b. E(X) = 5, V(X) = 1.33, probability is zero (0) because X has a uniform distribution O c. NONE O d. E(X) = 5.5, V(X) = 2.0833, probability is zero (0) because X has a uniform distribution
Calculate estimates of the standard deviations sz, Sy of the samples r = (5,9, 7) and y = (-1,2,5) as well as the Pearson coefficient of correlation rry of r and y. Which of the following answers are correct? a) sz = 4, sy = 9 b) rzy = 0 c) sz = 2, sy = 3 d) Tzy = } e) rzy =

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