3) Let Y,., Y, be iid Poisson(6). Find the UMP test of Ho : 0 < 8o versus H1 : 0 > 8o using the Neyman Pearson lemma. Your final test statistic should have a Poisson distribution when Ho is true.
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- Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θIf X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?
- Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.Let X1, ..., Xn be a sample from an Poisson population with parameter λ.(a) Find the maximum likelihood estimator for λ.(b) Is the estimator unbiased?(c) Is the estimator consistent?
- X is an poisson random variable with parameter λ = 4 Calculate P {X ≤ 3}A student attempted to use L’Hopital’s Rule as follows. Did the student make an error, if any, or does it state “no error”?Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ = 20 (suggested in article "Dynamic Ride Sharing: Theory and Practice," J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will: (i) Be at most 10? (ii) Exceed 20? (iii) Be between 10 and 20, inclusive? (iv) Be strictly between 10 and 20?
- Let X1 and X2 constitute a random sample from a nor-mal population with σ2 = 1. If the null hypothesis μ = μ0 is to be rejected in favor of the alternative hypothesis μ = μ1 > μ0 when x > μ0 + 1, what is the size of the criti-cal region?The contingency table shows the result of a random sample of students by Ziyi Zhang at PCC and the number of hours they spend investing on Robinhood and Acorns by age and gender. At alpha =0.05, can you conclude that age is related to gender when it comes to time spent investing on Robinhood and Acorns. GENDER 16-20yrs 21-30yrs 31-40yrs 41-50yrs 51-60yrs 61yrs or older Male 45 170 90 72 45 26 Female 9 30 21 17 10 5 What is the Chi-square test statistics and the decision for this question. The Test statistics is . Leave your answer in 3 decimal places. DECISION: We the null hypothesis.Let X1, X2, …, Xn be a random sample from the Normal distribution N() (a) Using method of moments to estimate the parameters and . (b) Are those estimators unbiased?