2.4 Consider the independent r.v.'s x,..,.X with the Weibull p.d.f. (i) Show that- is the MLE...
3.18 Let the r.v. X has the Geometric p.d.f. (i) Show that X is both sufficient and complete. U(X )-1 (ii) Show that the estimate U defined by: estimate of 6 if X-1, and U(X) -0 if X 2 2, is an unbiased (iii) Conclude that U is the UNU estimate of θ and also an entirely unreasonable estimate.
1.5 If Xi, ..., X, are independent r.v.'s distributed as B (k, θ), θ e Ω2(0.1), with respective observed values x, , xu, show that k is the MLE of θ, where x is the sample mean of the x's.
Consider X-B(3,1/4). The p.d.f of this discrete r.v. X is:
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).
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0. Compute the probability: P(X+Y > 2). Hint: Use independence of X and Y in order to find their joint p.d.f., fx,y, and then use the diagram below to compute the probability: P(X+Y < 2). y 2 r+y = 2 y . ! 2 0 2-y Note: If X and Y represent the lifetimes of 2 identical equipment of expected lifetime 1 time unit, then...
5.1 Refer to Exercise 1.6, and derive the moment estimate of θ. Also, compare it with the MLE θ= 1/x. 1.6 If the independent r.v.'s. Xi, .., X" have the Geometric p.d.f.
21 Let x.., X, be ii.d. r.v.s with the Negative Exponential p.d.f., /(x.6)0,ee-(O.) Then , x > 0,9 e Ω-(0,00 (i) Show that 1 / X is the MLE of θ.
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected lifetime. On the basis of the random sample Xi,..X from this distribution, derive the MP test for testing the hypothesis Ho:to against the alternative HA:-(>o) at level of significance o
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected...
P(8), θ eQa(0 4.6 Let X, ..., X, be independent r.v.'s distributed as estimate δ(x , , x )-r and the loss function L ( , δ)-[8-6(5- ,o0 ), and consider the -5)]218. E,[e-5(X, ,X,)],andshow that it isin dependent ofe. R(0:δ)- (i) Calculate the risk (ii) Can you conclude that the estimate is minimax by using Theorem 9?
5.7 Let X, X, be independent r.v.'s from the u(e -a, o+ b) distribution, where a and b are (known) positive constants and θ Ω M. Determine the moment estimate θ of θ, and compute its expectation and variance.