Concept explainers
Use the method described in Exercise 9.26 to show that, if Y(1) = min(Y1, Y2, . .., Yn) when Y1, Y2, … , Yn are independent uniform random variables on the interval (0, θ) , then Y(1) is not a consistent estimator for θ. [Hint: Based on the methods of Section 6.7 , Y(1) has the distribution
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Chapter 9 Solutions
Mathematical Statistics with Applications
- (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.arrow_forwardLet X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0arrow_forwardConsider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?arrow_forward
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- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.arrow_forwardLet X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.arrow_forwardSuppose a continuous random variable X has the following CDF:: F(x) = 1 - 1/ (x+1)4, x > 0.Find SX (x), survival function? a 1/(x+1)4, x < 0 b -1/(x+1)4, x > 0 c (x+1)4, x > 0 d 1/(x+1)4, x > 0arrow_forward
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