Statistics MATH 324 Assignment 2 complete solutions correct answers key

Statistics MATH 324 Assignment 2 complete solutions correct answers key

McGill University

Q9 Resistors to be used in a circuit have average resistance 200 ohms and standard deviation

10 ohms. Suppose 25 of these resistors are randomly selected to be used in a

circuit.

(a) What is the probability that the average resistance for the 25 resistors is between

199 and 202 ohms?

(b) Find the probability that the total resistance does not exceed 5100 ohms.

Q10 Suppose that

X1; : : : ;Xn; Y1; : : : ; Yn

are independent random samples from distributions with means _1 and _2, respectively,

and variances _2

1 and _2

2, respectively. Prove that

Un =

( _X

n 􀀀 _ Yn) 􀀀 (_1 􀀀 _2) p

(_2

1 + _2

2)=n

converges in distribution to the standard normal distribution as n ! 1.

Q11 According to a survey conducted by the American Bar Association, 1 in every 410

Americans is a lawyer, but 1 in every 64 residents of Washington, D.C., is a lawyer.

(a) If you select a random sample of 1,500 Americans, what is the approximate

probability that the sample contains at least one lawyer? Is n = 1500 large

enough for the approximation to work well?

(b) If the sample is selected from among the residents of Washington, D.C., what is

the approximate probability that the sample contains more than 30 lawyers? Is

n = 1500 large enough for the approximation to work well in this case?

(c) If you stand on a Washington, D.C., street corner and interview the first 1,000

persons who walked by and 30 say they are lawyers, does this suggest that the

density of lawyers passing the corner exceeds the density within the city? Explain.

Q12 The number of breakdowns per week for a type of minicomputer is a random variable

Y with a Poisson distribution with parameter _ > 0. A random sample Y1; : : : ; Yn of

observations on the weekly number of breakdowns is available.

(a) Suggest an unbiased estimator of _ and compute its MSE.

(b) Place a 2-standard-error bound on the error of estimation.

(c) The weekly cost of repairing these breakdowns is C = 3Y + Y 2. Show that

E(C) = 4_ + _2.

(d) Find an unbiased estimator of E(C) and assess whether it is consistent.

Johanna G. Nešlehová

Statistics MATH 324

McGill University, Fall Term 2015

Assignment 2: Due October 15 at 11:59 PM

Q13 Let X1; : : : ;Xn be a random sample of size n from a continuous distribution with DF

f(x; _) =

_

__ x_􀀀1; 0 _ x _ _

with parameter _ > 0. The value of _ > 0 is assumed to be known and fixed.

(a) Show that

T(X1; : : : ;Xn) = max(X1; : : : ;Xn)

is a biased estimator of _. Is it asymptotically unbiased?

(b) Find a multiple of T, say ^_n, that is an unbiased estimator of _.

(c) Show that

~_n =

_ + 1

_

_X

n

is also an unbiased estimator of _.

(d) Compute the efficiency of ^_n relative to ~_n and decide which estimator is better.

(e) Show that ^_n and ~_n are consistent.

Q14 Let Y1; : : : ; Yn be a random sample from a Pareto distribution with parameters _ > 0

and _ > 0. The density of this distribution is

f(yj_; _) =

(

___y􀀀(_+1); y _ _;

0; elsewhere:

(a) If _ is known, show that

Qn

i=1 Yi is sufficient for _.

(b) If _ is known, show that min(X1; : : : ;Xn) is sufficient for _.

(c) What is the sufficient statistic for the pair (_; _) in case both parameters are

unknown?

Q15 (optional) Let X1; : : : ;Xn be a random sample from a distribution with DF f(_; _) with unknown

parameter _. If ^_n is an unbiased estimator of _, then under very general conditions,

var(^_n) _ I(_) =

_

n E

_

􀀀

@2 ln f(X; _)

@_2

__􀀀1

This is known as the Cramer–Rao inequality. If var(^_n) = I(_), the estimator ^_n is

said to be efficient.

(a) Suppose that f is the normal density with mean _ and variance _2. Show that

_X

n is an efficient estimator of _.

(b) Suppose that f is the probability mass function of the Poisson distribution with

parameter _. Show that _X

n is an efficient estimator of _.