Let X1, ··· ,X n be i.i.d . random variables with probability density function,

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Let X1, ··· ,X n be i.i.d . random variables with probability density function,
fx (x ) = { lIJ 0 x 0
otherwi se.

  1. [6 marks] Let Xi, ··· , X denote a bootstrap sample and let X = 2: 1 x; .

Find: E(X IXl, . . . ‘ X n) , V(X IXl, . . . ‘ X n) , E(X ), V(X ).
Hint: Law of total expectation: E(X) = E(E(X IY)).
Law of total variance: V(X ) = E(V(X I Y)) + V(E(X IY)).

1

Sample variance, i.e. 82 = n 1

  1. [6 marks]

(Xi – X)2is an unbiased estimator of population variance.

A A

Let 0 = max(Xi, ··· , X n) and O* = max(Xi, ··· , X ) . Show as the sample size goes larger, n —+ oo,
A A 1
P( O* = 0) —+ 1- -.
e

  1. [6 marks]

Design a simulation study to show that (b)
A A 1
P( O* = 0) —+ 1- -.
e
Hint: For several sample size like n = 100, 250, 500, 1000, 2000, 5000, compute the approximation of
P( B* = 0).

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