For shifted exponential the
domain of x depends on the parameter so the likelihood function is
maximized at the sample minimum value
1.11 function p.d.f., , is a On the basis of a random sample of size n...
random sample of size n from the p.d.f. 1.8 On the basis of a (x,θ)-θΧθ-1 , 0 < x < 1, θ E Ω = (0,0)derive the MLE of θ
2.11 Refer to Exercise 1.11, and determine: (i) A sufficient statistic for a whenỗ is known (ii) Also, a sufficient statistic for β when a is known. (ii) A set of sufficient statistics for a and ß when they are both unknown. the function , is a p.d.f.,
A random sample of size n = 2 is taken from the p.d.f
f(x) = {1 for 0 ≤ x ≤ 1 and 0 otherwise.
Find P(X-bar ≥ 0.9)
3. A random sample of size n = 2 is taken from the p.d.f 1 for 0 < x < 1 f(30 0 otherwise. Find P(X > 0.9)
2.6 the function f(x θ)-6x-(θ+1), x 2 lde Ω-(0,0) is a p.d.f. (ii) On the basis of a random sample of size n from this pdf., show that the statistic X, sufficient for θ X is
QUESTION 5 Let Y , Y2, , Yn denote a random sample of size n from a population whose density is given by (a) Find the method of moments estimator for β given that α is known. Find the mean and variance of p (b) (c) show that β is a consistent estimator for β.
(10) For a random sample of size n from a Beta(α, β) density, find a consistent estimator of β . Why is this estimator consistent?
(10) For a random sample of size n from a Beta(α, β) density, find a consistent estimator of β . Why is this estimator consistent?
x, R(x), is defined as the probability that X>x; ie, R(x)-P(X> x). Now, suppose that X has the Negative Exponential p.d.f, Then (ii) Use Theorem 1 in order to determine the MLE of R(x; θ), on the basis of a random sample X1,…, X from the underlying p.d.f. Theorem 1 Let θ-θ(x) be the MLE of θ on the basis of the observed values x1, , xn of the random sample X1, , X, from the pdf f(", θ), θ...
(1 point) Let X1 and X2 be a random sample of size n= 2 from the exponential distribution with p.d.f. f(x) = 4e - 4x 0 < x < 0. Find the following: a) P(0.5 < X1 < 1.1,0.3 < X2 < 1.7) = b) E(X1(X2 – 0.5)2) =
2.3 Let X be a r.v. describing the lifetime of a certain equipment, and suppose that the p.d.f. of X is f (ii) We know (see Exercise 2.1) that the MLE of θ, based on a random sample of size n from the above pd.f., is θ = 1/ X. Then determine the MLE of g(9).
3. (10 points) Based on a random sample of size n from the pdf below, derive the MLE of o. , 00