8. Consider a random sample X1,..., X, from the density f(x | 0) = me-l|/0 for x E R. (a) Find the MLE ôô for 0.
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- 8. (p.158/# 4.67) Find the Mean of the Random Variable g(x, y) = +X2Y, where the Joint Density Function of X and Y is f(x,y) = (x+2y), 0<<1, 1< y < 2, elsewhere.*9.52 Let Y₁, Y₂, ..., Y, be a random sample from a population with density function 10-12 f(y|0) = 0 ≤ y ≤0, elsewhere. Show that Y() = max(Y₁, Y₂, ..., Y) is sufficient for 0.6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?
- 6.74 Let Y₁, ₂, ..., Y, be independent, uniformly distributed random variables on the interval [0, 0]. Find the a probability distribution function of Y() = max(Y₁; Y₂;; Y₁). b density function of y(n). c mean and variance of y(n).The probability density function of a random variable Y is given by 24y(10+2y-0.3y*) f; (y)=< , 0< y<10 10000 0, otherwise Suppose a random sample of 40 observations is obtained from this distribution and these are denoted as Y,,Y,,…,Y40• 40 a) Approximate the probability that the sample sum T =Ey, is between 200 and 250. i=1 b) Approximate the probability that the sample mean Y is less than 5.5.3. A machine produces spheres with a random radius R which follows a distribution with Scr*, 07.1 Let X be a random variable with probability Si. 1= 1,2, 3, S(2) =10 elsewhere. Find the probability distribution of the random vari- able Y = 2X 1. 7.8 A dealer's profit, in units of $5000, on a new au- tomobile is given by Y = X², where X variable having the density function sa random S(z) =2(1 - 2), 02. Let Y₁, Y2,..., Yn denote a random sample from the probability density function (0+1)yº; 0 −1 otherwise. f(y;0) = { 0, (a) Find the MLE for 0, say . Be sure to check the second derivative. (b) Find the method of moments estimator of 0, say . (c) Suppose that 1000 'yi = 750.7516 and 1000In(y) = -329.0056, calculate 8 and 0. i=19. Let X be a random variable with density function f and characteristic function p. Show, subject to an appropriate condition on f, that [ f(x)² dx = 12/16 (1) 1² dr. 276. The random variable X is distributed by f(z)=e, z>0. What is the density, g(y), of where it is non-zero? A. 1 X²¹ 2y³/2 B. 2√9 C. 29-√7 D). -1/47 -1/ E. eThe probability density function of a random variable Y is given by 24y(10+2y- 0.3y²) S; (y)=< 0 < y<10 10000 0, otherwise Suppose a random sample of 40 observations is obtained from this distribution and these are denoted as Y,,Y,,…,Y40 • 40 a) Approximate the probability that the sample sum T =Ey, is between 200 and 250. i=1 b) Approximate the probability that the sample mean Y is less than 5.5.46. Given that X has a density function 2x 3- x f (x) = for 0SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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