2. (Modified Question 6.5.2. in page 382 of the textbook) Let Y,.., Yio be a random sample from a...
Question 1: (10 marks) Let Y, Y....,Y, be a random sample from the beta distribution with a = B = 4, and I2 = { u u = 1,2). Write the likelihood ratio test statistic A for testing Ho : H = 1 versus H:u= 2. Note that the pdf of a beta(a,b) distribution is as follows: com_(a+b)/2-1(1 - 0)8-1, 0<I<1. f(x) = f(a)(B)"
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider the hypotheses Hop-po against HA:p#po- a. Find the likelihood function of p in terms of random variable Y, L(p). b. Construct the (generalized) likelihood ratio λ(v). Hint: what is pMLE?] C. (i) For the particular case of po 0.25 and n 5, fill in the table: 3 4 A(y) (ii) Rearrange the table in the order of increasing of values of 2, and compute cumulative...
B1. A random sample of n observations, Yi, ., Yn, is selected from a pop- ulation in which Yi, for i = 1, 2, ,n, possesses a common distribution the same as that of the population distribution Y (a) Suppose that we know Y has a Geometric distribution with parameter p, p unknown. Find the estimator using the method of moments. (b) Suppose that we know that Y has an exponential distribution with parameter λ, λ unknown. Find the estimator...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
Let X1,...............,Xn be a sample from an exponential population with parameter λ. (a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased? (c) Is the estimator consistent?
2. (20pts) Let Xi,..., X be a random sample from a population with pdf f(x)--(1 , where θ > 0 and x > 1. (a) Carry out the likelihood ratio tests of Ho : θ-a, versus Hi : θ a-show that the likelihod ratio statistic corresponding to this test, A, can be re-written as Λ = cYne-ouY, where Y Σ:.. In (X), and the constant c depends on n and θο but not on Y. (b) Make a sketch of...
Exercise 3. (QUANTIZATION, FROM TEXTBoOK, PROBLEM 4.61) Let X be an exponential random variable with parameter λ. (a) For some d 〉 0 and k a nonnegative integer, find P(kd 〈 X 〈 (k + 1)d) (b) Segment the positive real line into 4 equally probable disjoint intervals.
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
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Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....