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5. Let X be the number of calls received during any one hour, and follow a Poisson distribution with puf: P(X = ) =

(d) (2 points) When , reject H₀. Find the power of the test.

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1. (10 points) Let X~Beta(a,β) and Y~Beta(a + β,γ) be two independent variables of beta distribution. Find the distribution of XY by making the transformations U=XY, V=Y. The pif of beta distribution, Beta(a,b), is given by

(a) (5 points) Find the MLE (maximum likelihood estimator) of 6. Is it an unbiased estimator?

(b) (10 points) Use the MLE in (a) to obtain a 100 x (1 - c)% confidence interval for θ. Hint: is distributed as a Chi-square distribution.

(a) (5 points) Find the method of moment estimator of θ.

(b) (10 points) Find the limiting distribution of as n→∞ by the Delta method, where Tis the method of moment estimator of θ.

(a) (5 points) Find the uniform minimum variance unbiased estimator (UMVUE) of g(μ).

(b) (5 points) Find the Cramer-Rao lower bound (CRLB) for the variance of unbiased estimator of g (μ). Is the CRLB attained by the variance of the UMVUE?

(a) (2 points) When significant level is set as 0.05, fine the rejection region of X.

(b) (2 points) When X, reject H0. Find the probability of type I error.

(c) (2 points) When X , reject H0. Find the probability of typeerror.

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110年 - 110 國立臺灣大學_碩士班招生考試_農藝研究所生物統計學甲組:統計理論#102885

110年 · #102885

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