X, be a random sample from a distribution with the probability density function f(x; θ) = (1/02).re-z/. 0 <エく00, 0 < θ < oo. Find the MLE θ
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞.
(1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show
that this MLE is unbiased.
Exactly 6.4-8. Let f(x:0)-缸붕 for 0 < x < 1,0 < θ < oo 1 1-0 (1) Show that θ Σ-1 log(X) is the MLE of θ (2) Show that this MLE is unbiased.
Exactly 6.4-8. Let f(x; θ)-29 for 0 < z < 1,0 < θ < oo. (1) Show that this MIE is unbiased the MLEofe. 1-0 (2) Show that this MLE is unbiased.
Let Xi, X2, Xn be ar ensity function f(r; θ) = (1/2)e-Iz-이,-oo < x < 00,-00 < θ < oo. Find the d MLE θ
2. Let Xi, X2, . Xn be a random sample from a distribution with the probability density function f(x; θ-829-1, 0 < x < 1,0 < θ < oo. Find the MLE θ
C3.)Let f(z,0) (1/θ)2(1-0)/0, 0 < x < 1,0 < θ < oo. . Find the maximurn likelihood estimator of θ. Show that the maximurn likelihood estimator is unbiased to θ
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
Let X have probability density function f(2)= k(1+x) -3 for 0 < x < oo and f(x) = 0 elsewhere. a. Find the constant k and Find the c.d.f. of X. b. Find the expected value and the variance of X. Are both well defined? c. Suppose you are required to generate a random variable X with the probability density function f(x). You have available to you a computer program that will generate a random variable U having a U[0,...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
7. For the probability density function f(x) = for 0 <<<2 (a) Find P(x < 1) (b) Find the expected value. (c) Find the variance.