Find the Fisher information in this sample of size n about the parameter 8.
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- 25. 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.Suppose that Y1, . . . , Yn is a random sample from a population whose density function isLet X₁, X2,..., X₂, be a random sample from a population with cumulative distribution function F(x) = xº, 0 0.
- Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 froma distribution having the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Show thatZ1 = Y1/Y2, Z2 = Y2/Y3, and Z3 = Y3 are mutually independent.Let X be the random variable having pdf f(x) = 1,0 < x < 1, zero e.w. Compute the probability of the range of the random sample of size 4 (X1, ., X4) is less than 1/2. Hint: Range is R = X(4) – X(1).Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?
- Let X1, X2, ..., X50 be a random sample of size 50 from a distribution with density for 0Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?Let X,, X2,... ,X, be a random sample from a distribution with density function if x20 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?Let Y₁, Y₂, . density function Yn denote a random sample from a uniform distribution with " 0 ≤ y ≤0, 0, elsewhere. For testing Ho: 0 = 0o vs. Ha: 000, show that the a-level Likelihood Ratio Test (LRT) gives RR = {Y(n) > 00} U {Y(n) < 00α = } . f(y|0) =Let X1, X2, ..,X, be a random sample of size n from a distribution with probability density function f(x; 8) = {o." 0 : ( 0(1+x)-(1+0), x>0 elsewhere Find the Maximum likelihood estimator (MLE) of 0.Let X1,., X, denote a random sample from the probability density function f(r) = 0xº-1, 0 0. Show that X is a consistent estimator of .
