Math 494 - Homework

# Homework

## Practice problems (not due)

Please consider the following problems from the textbook:

Hogg-McKean-Craig, p339: 6.4.3, 6.4.10 (ignore the asymptotically independence part)

Also: relate the result of 6.4.10 with the 2-parameter Fisher information for the normal distribution, as worked out in class.

## Homework 10, due Apr 29 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p339: 6.3.5, 6.3.6, 6.3.9, 6.3.10, 6.3.15, 6.3.18

A.  Using Theorem 10.2.3 (as discussed in class), devise a chi-squared hypothesis test based on the median M of a location model. State carefully what assumptions or regularity conditions you need.

On 6.3.6, you can skip the computational/power curve part.

## Homework 9, due Apr 20 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p330: 6.2.5, 6.2.6a

A.  Prove that if X is any continuous random variable (not necessarily having an expected value), then E( X2 / (1 + X2) ) necessarily exists.

B.  Explain in detail how to set up a Wald test for a null hypothesis of Theta0 in Exercise 6.2.14 at the 0.05 level. Please fit to a chi-squared distribution.

## Homework 8, due Apr 13 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p330: 6.2.3, 6.2.4, 6.2.8, 6.2.11, 6.2.14
Hogg-McKean-Craig, p207: 4.2.1

Hint 1: On 6.2.3, you'll need to do inverse tangent substitution. It may be easier to work the integral arising from the square of the first derivative, rather than the second derivative. (Alternatively, on an integral of this difficulty, there is no shame in using Wolfram Alpha or similar.)
Hint 2: On 6.2.11, one approach to finding the 4th moment of a normal random variable would be to look at the mgf. Try thinking of it as the product of the mgf for \mu and for a normal r.v. with mean 0.

## Homework 7, due Apr 1 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p330: 6.2.1, 6.2.2, 6.2.7bc, 6.2.9, 6.2.10

## Homework 6, due Mar 25 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p317: 6.1.2, 6.1.3, 6.1.6, 6.1.11, 6.1.13
Hogg-McKean-Craig, p331: 6.2.7a

A.  Prove Jensen's inequality (non-strict version) for a geometric random variable X with p = 1/2, and any bounded convex continuous function \phi. (Bounded means that |\phi(t)| < c for some fixed constant c and any t.)
Hint: Write your infinite sums as the limit of expected values of finite probability spaces, and apply Part A from last week.

## Homework 5, due Mar 11 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p317: 6.1.1, 6.1.4, 6.1.5, 6.1.7, 6.1.10

A.  Prove Jensen's inequality (non-strict version) for a discrete random variable X with finite support, i.e., X takes on real values a1, ..., an, with p(ai) = pi.
Hint: induct on n, using conditional probability.

B.  Prove Jensen's inequality (non-strict version, where \phi is any convex continuous function) for a random variable with continuous pdf f supported on (a, b), where a and b are real numbers (i.e., not infinite).
Hint: use the Riemann integral definition, and apply Problem A. The Mean Value Theorem for Integrals may be helpful to get a pmf in your Riemann integral.

## Homework 4, due Mar 2 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p246: 5.2.1 (exponential only), 5.2.7, 5.2.9, 5.2.15, 5.2.27
Hogg-McKean-Craig, p261: 5.4.13
Hogg-McKean-Craig, p285: 5.7.7

A.  Create (by hand) the normal qq-plot for the following sample: 1, 2, 4, 7.

Note that the beta pdf (as referenced in 5.2.9) can be found in Chapter 3.3.

## Homework 3, due Feb 16 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p170: 3.4.26
Hogg-McKean-Craig, p188: 3.6.6, 3.6.10
Hogg-McKean-Craig, p201: 4.1.1, 4.1.2 (also compute E(Yn) ), 4.1.8, 4.1.26
Hogg-McKean-Craig, p238: 5.1.4abc

Remark.  Please note that 4.1.26 refers to 4.1.25. The result of 4.1.25 is that, although S2 is an unbiased estimator for variance, S is not an unbiased estimator for standard deviation.

## Homework 2, due Feb 9 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig, p158: 3.3.17, 3.3.26ab
Hogg-McKean-Craig, p170: 3.4.21
Hogg-McKean-Craig, p201: 4.1.6 (variance only), 4.1.12, 4.1.22

A.  Let T be a random variable with a t-distribution having 1 degree of freedom. Using the pdf directly (without using a computer), find P( 0 < T < 1).
Please be careful to show an appropriate level of work on this problem.

B.  Let Z be a standard normal random variable.
i.  Without using Theorem 3.4.1 or any of its consequences, find all moments E( (Z2)k ) of Z2.
ii.  Use moment generating functions and part (i) to prove Theorem 3.4.1.

## Homework 1, due Feb 4 at 2:07pm

Please do the following problems from the textbook:

Hogg-McKean-Craig,   p65: 1.9.7, 1.9.8, 1.9.18
Hogg-McKean-Craig, p115: 2.5.13
Hogg-McKean-Craig, p140: 3.1.2, 3.1.24
Hogg-McKean-Craig, p157: 3.3.1
Hogg-McKean-Craig, p170: 3.4.29