14.1 (w, f) If Y= X₁ + X₂ + X3, where X~ N(μ, C) and 1 [ | | -₁ = { 1 2 3 μl = find the mean and variance of Y. 1 1/2 1/4 1/2 1/4 1 1/2 1/2 1
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- a. Show that the regression R2 in the regression of Y on X is the squaredvalue of the sample correlation between X and Y. That is, show thatR2 = r2XY.b. Show that the R2 from the regression of Y on X is the same as the R2from the regression of X on Y. c. Show that ^β1 = rXY (sY/sX), where rXY is the sample correlationbetween X and Y, and sX and sY are the sample standard deviationsof X and Y.Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
- 4 Let f(x) = (1/10)(x-3)^2 for x = 1, 2, 3, 4, 5 %3D Find the standard deviation of X. For the instructor, this was question 11. / 1.8 X 3.1 X 3.4 X 3.5 X 9.4 X 12.4If X ~B(190,0.34), find mean and variance of x. (Rounded in 3dp.)A professor grades on a curve by assigning C's to all scores from u- ; to u+%, B's to all scores from i+% to µ+ , and D's to all scores from u – * to µ – 5. Everyone who scores below a D gets an F, and everyone who scores above a B gets an A. a. If the scores are normally distributed with mean u and variance o², what percentage of students will receive each grade? Hint: convert to z-scores and use pnorm() to find probabilities (percentages) b. If the scores are uniformly distributed (continuous) from 0 to 100 what percentage will receive each grade? Hint: first determine the values of u and o for this distribution
- F. Suppose that a random variable X has normal distribution with mean µ = 2 and variance σ 2 = 9, that is, X ∼ N(2, 9). (30) E[(X + 2)^2 ] is (a) 20 (b) 25 (c) 15 (d) 30 (31) The variance of X/2 + 3 is (a) 9/4 (b) 3/8 (c) 9 (d) 9/2Suppose that f (x) = x / 8 for 3 < x < 5. Determine the mean and variance of x.4. If x' and y' are the deviations of the variables X and Y from their means, s, s are the variances of variables X and Y, correlation coefficient between variables X and Y is r and (X, Y) for i = 1, 2, n are the n values of (X, Y), then show that Hence prowa x x $2 $1 =1-12/12/2012 2n ...9 100 =−1+ Xi Xi + 2n 2n=1 $1. $₂ balles ad no
- Suppose X, Y, Z are i.i.d. from a Normal distribution with mean μ and variance 4. You have to compare between three estimators for μ, which are T1:= (2X + Y)/3 , T2:= (X + Y − Z)/2 , T3:= (Y + 2Z)/2 . (a) Which among the above are unbiased estimators for μ? (b) Which among these have the smallest variance? (c) Can you propose an estimator for μ which is better than the 3 above? You have to show that your estimator is indeed better.4Q1. (3 points) Suppose that samples of size 36 are selected at random from a normal population with mean 78 and variance 49. What is the probability that the sample mean falls in the interval from Ax- 1.40g and ux +2.4og.