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Let Y1, Y2, … , Yn denote a random sample of size n from a uniform distribution on the interval (0, θ). If Y(1) = min(Y1, Y2, … , Yn), the result of Exercise 8.18 is that
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Chapter 9 Solutions
Mathematical Statistics with Applications
- Assume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .arrow_forwardLet f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isarrow_forwardFind the minimum mean square error forecast Y(1), forecast error e, (1) and Varfe, (1)1 for the following modes. Y, = 0.8Y, +e,. Y, = 3+21+e,.arrow_forward
- EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.arrow_forward(13) Let X b(6,-) find E(5+6x) and distribution function. 3arrow_forwardThe value of the correlation r = (x,y) for the information would be: n = 8, Sum(x) = 609, Sum(y) = 504, Sum (x)^2 = 47995, Sum (y)^2 = 32954, and Sum (xy) = 395650 0.255 0.147 0.855 0.174arrow_forward
- B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.arrow_forwardThank youarrow_forwardConsider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.arrow_forward
- 2) Let X be a continuous random variable with PDF 0arrow_forward25. Obtain the maximum likelihood estimates for a and ß in terms of the sample observations x1, x2, X taken from the exponential population with density functions : f (x, a, B) = yoe-B(*- ), a0 ", asx0 Y'o being the constant.arrow_forwardA variable X has a probability function f(x, 0) = 0e 0x; x≥0 y >0. The parameter 0 is estimated using two different estimators of simple random samples of size two: 01 2x+4x2 and 2=4x+5x2 3 3 Select the best estimator for \theta by comparing the Mean Squared Errorsarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage