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 Theorem, what is the approximate distribution for when n is large?
(e) Suppose that = 5.34 for a random sample of size 64, determine the approximate 95% confidence interval for µ?
Please show as many steps as possible because I am struggling with this class right now and I am not understanding the estimation methods very well. PLEASE HELP ME!
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the density f(x) = 1 µ e −x/µ for x > 0. (a) Let ˆµ1 = X. Show ˆµ1 is a minimum variance unbiased estimator. (b) Let ˆµ2 = (X1 +X2)/2. Show ˆµ2 is unbiased. Calculate V ar(ˆµ2). Confirm V ar(ˆµ1) < V ar(ˆµ2). Calculate the efficiency of ˆµ2 relative to ˆµ1. (c) Show X is consistent and sufficient. (d) Show ˆµ2 is not consistent...
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Let X1, ..., Xn be a random sample (i.i.d.) from a normal distribution with parameters µ, σ2 . (a) Find the maximum likelihood estimation of µ and σ 2 . (b) Compare your mle of µ and σ 2 with sample mean and sample variance. Are they the same?
Let X1, . . . , Xn be a random sample from a triangular probability distribution whose density function and moments are: fX(x) = * I{0 x b} a. Find the mean µ of this probability distribution. b. Find the Method Of Moments estimator µ(hat) of µ. c. Is µ(hat) unbiased? d. Find the Median of this probability distribution. I will thumbs up any portion or details of how to do this problem, thanks! We were unable to transcribe this...
A random variable X has probability density function f(x)=(a-1)x^(-a),for x>=1. (a) For independent observations x1,...,xn show that the log-likelihood is given by, l(a;x1,...,xn)=nlog(a-1)-a (b) Hence derive an expression for the maximum likelihood estimate for ↵. (c) Suppose we observe data such that n = 6 and 6 i=1 log(xi) = 12. Show that the associated maximum likelihood estimate for ↵ is given by aˆ ↵ =1 .5. logri We were unable to transcribe this image
Let X1, X2, ..., Xn be a random sample of size n from the distribution with probability density function To answer this question, enter you answer as a formula. In addition to the usual guidelines, two more instructions for this problem only : write as single variable p and as m. and these can be used as inputs of functions as usual variables e.g log(p), m^2, exp(m) etc. Remember p represents the product of s only, but will not work...
X1, X2, . . . , Xn i.i.d. ∼ N (µ, σ2 ). Assume µ is known; show that ˆθ = 1 n Pn i=1(Xi− µ) 2 is the MLE for σ 2 and show that it is unbiased. Exactly 6.4-2. Xi, X2, . . . , xn i d. N(μ, μ)2 is the MLE for σ2 and show that it is unbiased. r'). Assume μ is known; show that θ- n Ση! (X,-
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