!If x andy are stochastically independent r.v.s= then the mean of x and y (E(xy)) isand E(x) = 4, E(y) =Option 1Option 3|||121U

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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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If X is a random variable with expectation µ and variance cµ2 , where c is a constant. Find a variance stabilizing transformation of X.
Suppose that approximately 40% of the population has Type O+ blood. You take a sample of 5 persons. Let Y denote the number of persons in the sample with Type O+ blood. Find each of the following. (a) Pr{Y = 2} = (b) Pr{Y < 2} = (c) Pr{Y ≥ 2} =
x = y + 3 3y + 2x = 16 5y = 13 y = 2.6 noit op bit x = y + 3 = 2.6 + 3 x = 5.6 eew 3(y+3) + 2y = 16 3y +3 + 2y = 16 5y +3 = 16 %3D oy 10ode bo bot
Exercise 4. (a) x²(5.7) ~ Gamma(?,?), (b) x²(8) ~ Gamma(?,?), (c) - Gamma(4, 3) ~ x² (?)
1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
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.
Please no written by hand solution
7. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: The marginal function of x %3D
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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.
8. (a) Assume process X, satisfies X,-aX-1+ Z where Z, is WN(0, a?) and Ja< 1. Use the Yule-Walker equation to find the best linear predictor P(X+iXn Xn-1). Find the mean-square error of the predictor. = Z, + aZ,-1 + bZ,-2, where Z, is WN(0, a). Use the (b) Assume process X, satisfies X, Durbin-Levinson algorithm to find the linear predictor P(X+i|X Xn-1)-

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If x andy are stochastically independent r.v.³ and E(x) = 4, E(y) = then the mean of x and y (E(xy)) is 1 о Option 1 о Option 3 1