Concept explainers
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- Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.arrow_forwardX,X, and X, is a random sample of size 3 from a population 3. with mean value u and variance o2, T, T, T, are the 3. estimators used to estimate mean value u, where T=X,+X,-X,T,= 2X,+3X, - 4X,& T, ax,+x,+x) 1 (ax,+X,+X,) (i) Are T, and T, unbiased estimators ? 1 2 (ii) Find the value of such that T, is unbiased estimator 3 for u. With this value of A is T, a consistent estimator ? (iii) 3 (iv) Which is the best estimator ?arrow_forwardLet X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.arrow_forward
- Let ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.arrow_forwardLet x, x2, ..., Xn be a random sample from a Poisson distribution with parameter 2. Find ML estimator of 2. Also find its variance.arrow_forwardLet X be a discrete random variable with range Rx = {0, ,}, such that Px(0) = Px() = Px() = Px() = Px(*) = . Find Elsin(X). %3Darrow_forward
- Let X1, . X,, be n independent random variables with PDF laje- if x > 0, 10 fx, (x) = i = 1,2, ...,n. if x, 0 V i. Find the distribution of Y = min{X1, X2,..., X,) and Z = max{X1, X2,..., X„}.arrow_forward2) Let X be a continuous random variable with PDF 0arrow_forwardQ3) 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 0arrow_forward17. 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 maximumarrow_forwardSuppose 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 integerarrow_forwardLet X, X2, ... , X, be a random sample from a population that is uniformly distributed over (a, b), a, b e R, a < b. Let x be the sample mean and Y, = min {X; : i = 1, ..., n}. • Find the E[Y,).arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,