Show that there is a unique choice of L and y with E = LL' + p,%3D0.4 0.9]0.7 with onefor the population covariance matrix E = |0.4Lo.9 0.7%3D1factor m = 1

Answer: The model assumes that the data is a linear function of the common factors. However, because…

Answered: Show that there is a unique choice of L…

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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House values in Martin, TN are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that house values in Martin, TN have a mean of $64,500 with a standard deviation of $21,350. Answer the following probability questions concerning house values in Martin, TN. Part 1: If 1 randomly selected house in Martin, TN is selected, what is the probability that this house has a value under $60,000? Part 2: If 30 randomly selected houses in Martin, TN are selected, use the Central Limit Theorem to determine the probability that their average value is under $60,000?
Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.
For two variables X and Y, we have two regression lines 2X-7Y+3=0 and aX+bY+6=0 for some constants a and b. Suppose we know that X = 2 and Y = 1. Also the second regression line passes through (4, 5). Find the correlation coefficient of these two lines.
Question 6 Suppose A and B are two events with probabilities: Find the following: a) P(ANB). b) P(A). c) P(Bº). d) P((An B)*). Edit View Insert Format Tools Table 12pt v P(Aª) = .40, P(B) = .25, P(A U B) = .75. ✓ Paragraph BIUAT² : B Q Search V
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
2. Assuming the dichotomous disease model, with Y1 and Y2 being two relatives with relatedness R, show that (a) Cov(Y1, Y2) = P(Y1 = Y2 = 1) – K². (b) express R as a function of the covariance and K.
(b) Compute the covariance matrix of the data given by x1 = [2 1]', x2 = [3 2]', x3 = [2 3]'and x4 = [1 4]'.
2. (a) When two random variables (X, Y) follow a bivariate normal distribution, the covariance matrix Σ is defined as Σ ρσισ2 (010₂ Posts), - = where p is the correlation, o, o are the variances of X and Y respectively. Use mvrnorm from the MASS library (use library(MASS)) to generate 100 samples from a bivariate normal Call the first column x distribution 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 (use cor) and compare with the population correlation p. (First work out p in the Σ given.)
5.

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Show that there is a unique choice of L and with E = LL'+ 4, 1 for the population covariance matrix E = |0.4 Lo.9 0.4 0.9] 0.7 with one 1 0.7 1 factor m = 1 %3D