Suppose that X1, X, ..., Xn are a sequence of i.i.d. Exponential() variables. Use the delta...
Suppose that X1,... , X are i.i.d. with an exponential distribution with mean equal to 2.0. Assume that n = 40. Let X, = 1-1Xi. approximate distribution for 1/Xn, using the delta method Obtain an
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
Suppose X1, ?2, ... , ?? are i.i.d. exponential random variables with mean ?. a. Find the Fisher information ?(?) b. Find CRLB. c. Find sufficient statistic for ?. d. Show that ?̂ = ?1 is unbiased, and use Rao − Blackwellization to construct MVUE for ?.
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...
Suppose that the random variables X,..Xn are i.i.d. random variables, each uniform on the interval [0, 1]. Let Y1 = min(X1, ,X, and Yn = mar(X1,-..,X,H a. Show that Fri (y) = P(Ks y)-1-(1-Fri (y))". b. Show that and Fh(y) = P(, y) = (1-Fy(y))". c. Using the results from (a) and (b) and the fact that Fy (y)-y by property of uniform distribution on [0, 11, find EMI and EIYn]
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
Problem 3: Suppose X1, X2, is a sequence of i.i.d. random variables having the Poisson distribution with mean λ. Let A,-X, (a) Is λη an unbiased estimator of λ? Explain your answer. (b) Is in a consistent estimator of A? Explain your answer 72
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...