2. Let X₁, X, be a random sample from the Pareto distribution with the p.d.f.: f(x0) = (0/x²) 1(x>0), 0<<∞.(20) a) Show that X(1) min (X) is a CSS(Complete Sufficient Statistic) for 0.(10) (hint: Y= X(₁)~p(Y 0 ) b) Find the MVUE of 0.(10) (hint: X, = 0U₂, U₁ ~ƒ(u) = (1/u² ) 1.0) (u), U(1) ~ g(u) = n/u"+¹,1
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- 7. Let X be a continuous random variable with pdf f(x)=√ 03. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?17. Let (yı, Y2, Yn) be a random sample from a normal distribution such that *** E(y;) = a + Bxi, Var(y;) =o² where ri, i = 1,2, ..., n are constants, a, likelihood estimators of a , B and o?. B and o? are the parameters of interest. Find the maximumLet X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)F. Suppose that a random variable X has normal distribution with mean µ = 2 and variance σ 2 = 9, that is, X ∼ N(2, 9). (30) E[(X + 2)^2 ] is (a) 20 (b) 25 (c) 15 (d) 30 (31) The variance of X/2 + 3 is (a) 9/4 (b) 3/8 (c) 9 (d) 9/23. Let X1, X2, ..., Xn be a random sample from a distribution with pdf { r(4)04 Xª-'e¯3, 0Let X1,...,Xn be a random sample from the distribution f(x) = 2x, 0 < x < 1. Find the distribution of the sample maximum X(n)Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed.(a) Prove that the kth-order statistic, Y(k), has Beta distribution with α = k and β = n−k+ 1.(b) What is the distribution of the median of Y1,Y2,··· ,Yn when n is an odd integer?(c) Find the joint density of the middle two of Y(1),Y(2),··· ,Y(n) when n is an even integerQ3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 07. Let (r1, 12 ., T„) be a random sample from a normal distribution with mean u and a variance o?. Also let S (x; - )² be the sum of squares of the random sample. %3D State without proof the sampling distribution of the following; (a) t(x) = ao + b;a;, i = 0,1, .., n, a, are positive real constants. i=l n(ī – µ)² (x – 4) (b) t(r) = g2 (f) t(x) = (c) t(x) = (x – 4) (g) t(x) = i=1 (c) t(x)= i=1 i=1 (d) t(r) = E (*; – äm-n)² %3D j=1 i=m+1 (h) t(x) = m - 1 n - m - 1 (x – 4) (e) t(r) =Z is the present-value random variable for a whole life insurance of b payable at the moment of death of (x). You are given: (i) (ii) 8 = 0.05 (iii) The net single premium for this insurance is equal to Var(Z). Calculate b. x+t = 0.01 t≥0 (A) 1.36 (B) 1.68 (C) 2.00 (D) 2.32 (E) 2.64SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,