Let X1, , Xn be conditionally i i d with the uniform distribution on the interval 0, Let Yn max X1, , Xn a Find the p d f and the quantile function of Yn b Yn is often used as an estimator of even though it has bias Compute the bias of Yn as an estimator of c Prove that Yn is a pivotal d Find a confidence interval for with coefficient

Question: Let X1, . . . , Xn be conditionally i.i.d. with the uniform distribution on the interval [0, ]. Let Yn = max{X1, . .

Let X1, . . . , Xn be conditionally i.i.d. with the uniform on the interval [0, θ]. Let Yn = max{X1, . . . , Xn}.
a. Find the p.d.f. and the quantile function of Yn/θ.
b. Yn is often used as an estimator of θ even though it has bias. Compute the bias of Yn as an estimator of θ.
c. Prove that Yn/θ is a pivotal.
d. Find a confidence interval for θ with coefficient γ.

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