4. Let X1, X2, ..., Xn be a random sample from an Exponential(1) distribution. (a) Find...
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
Let X1, X2, . . . , Xn be a random sample from some distribution and suppose Y = T(X1, X2, . . . , Xn) is a statistic. Suppose the sampling distribution of Y has PDF fY (y) = 3 8 y 2 for 0 ≤ y ≤ 2. Find P[0 ≤ Y ≤ 1 5 ].
7. Suppose X1, X2, ..., Xn is a random sample from an exponential distribution with parameter K. (Remember f(x;2) = 2e-Ax is the pdf for the exponential dist”.) a) Find the likelihood function, L(X1, X2, ..., Xn). b) Find the log-likelihood function, b = log L. c) Find dl/d, set the result = 0 and solve for 2.
4. Let X1, X2, ..., Xn be a random sample from a distribution with the probability density function f(x; θ) = (1/2)e-11-01, o < x < oo,-oo < θ < oo. Find the NILE θ.
4. Let X1, X2, ..., Xn be a random sample from a distribution with the probability density function f(x; θ) = (1/2)e-11-01, o < x < oo,-oo < θ < oo. Find the NILE θ.
7.6.4. Let X1, X2,... , Xn be a random sample from a uniform (0,) distribution. Continuing with Example 7.6.2, find the MVUEs for the following functions of (a) g(0)-?2, i.e., the variance of the distribution (b) g(0)- , i.e., the pdf of the distribution C) or t real, g(9)- , î.?., the mgf of the distribution. Example 7.6.2. Suppose X1, X2,... , Xn are iid random variables with the com- mon uniform (0,0) distribution. Let Yn - max{X1, X2,... ,...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
5. Let X1, X2, ..., Xn be a random sample from a distribution with pdf of f(x) = (@+1)xº,0<x<1. a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for @ ?
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?
1. Suppose that X1, X2,..., X, is a random sample from an Exponential distribution with the following pdf f(x) = 6, x>0. Let X (1) = min{X1, X2, ... , Xn}. Consider the following two estimators for 0: 0 =nX) and 6, =Ỹ. (a) Show that ő, is an unbiased estimator of 0. (b) Find the relative efficiency of ô, to ô2.