STA452H1

UNIVERSITY OF TORONTO Faculty of Arts and Science

APRIL EXAMINATIONS 2021

STA452H1

(©David Brenner, 2021)

from WED, APR 16, 2021, 2:00pm

to THURS, APR 17, 2021, 2:00pm

24 hours: open book

Instructions

Please show all your work clearly in the space provided.

Note also that for this test there is the stipulated requirement

that you sign a contract to academic integrity both prior to and following completion of the test. For this please refer to next

page as well as final page.

Answer all five questions. All eight pages required. All complete questions will be valued equally at (20) each. Partial grades are shown in brackets to the right of each part

but partial credit will be awarded within each such part.

Q1 Q2 Q3 Q4 Q5 total

Name Student Number

REQUIRED: AGREEMENT TO ACADEMIC INTEGRITY 1

Academic integrity is a fundamental value of learning and scholarship at the University of Toronto.

Participating honestly, respectfully, responsibly, and fairly in this academic community ensures that

your UofT degree is valued and respected as a true signifier of your individual academic achievement.

Prior to beginning this term test, you must attest that you will follow the Code of Behaviour on

Academic Matters and will not commit academic misconduct in the completion of this test.

A�rm your agreement to this by completing the following Statement:

By signing this Statement, I, (print your full name here),

(student number here),

agree fully to abide by the Code of Behaviour on Academic Matters. I will not commit aca-

demic misconduct, and I am aware of the penalties that may be imposed if I commit an academic

o↵ence.

The University of Toronto’s Code of Behaviour on Academic Matters outlines such behaviours that

constitute academic misconduct, the processes for addressing academic o↵ences, and the penalties

that may be imposed. You are expected to be familiar with the contents of this document.

Potential o↵ences include, but are not limited to:

Using any unauthorized aid, including a cell phone, or searching for answers online.

Looking at someone else’s answers,

Letting someone else look at your answers.

Working together to answer questions.

Sharing the exam questions or discussing answers with anyone else.

Misrepresenting your identity or having someone else complete your test or exam.

signed

You may now proceed.

1. gamma basics: given X ⇠ gamma (p), p > 0 , Y ⇠ gamma (q), q > 0 and X ?? Y with U = X/(X+Y ) and V = X/Y , determine the following.

a) If EV = 1 and �(V ) = 2 what are each of p and q.

(7)

b) For the p and q determined above, obtain EU and �(U).

(5)

c) Thus find ⇢(U, V ).

(8)

2. the many perils of mean square error:

Given a sample X1, . . . , Xn IID X ⇠ N(µ, �2) for n � 2, let, as usual

nbµ = nX = n ⌃ i=1

Xi

& nb�2 = (n�1)s2X = n ⌃ i=1

(Xi�X)2 = n ⌃ i=1

X2i � nX 2 .

a) For (the square of) the L2-distance between the natural estimator, b�2, and the parameter, �2 : E(b�2��2)2 = an�4, determine an, n 2 N.

(5)

b) Do the same thing for the so-called unbiased estimator s2X : E(s 2 x��2)2 = bn�4.

(4)

c) Thus, by explicit di↵erence, determine which of the two is actually closer to �2.

(4)

d) It begs the question that there might well be an even closer estimator, cns 2 X :

(7)

3. rotational invariance and correlation: Y ⇠ unif (Bn) and U ⇠ unif (Sn�1)

a) With Xi = U 2 i , determine EX1, �(X1) and ⇢(X1, X2).

(10)

b) With Vi = Y 2 i , determine EV1, �(V1) and ⇢(V1, V2).

(10)

4. reciprocating gammas & the multivariate beta: Suppose that Zi indep ⇠ G(pi), i 2 N

and, for each n 2 N, set Sn = Z1+ · · · +Zn and Yn = Sn/Sn+1.

a) Prove that the Yn’s are mutually statistically independent.

(10)

b) Provide a joint probability density function for (Y1, . . . , Yn).

(5)

c) For pi = 1, i 2 N and Xn = Y nn , n 2 N obtain P(X1  · · ·  Xn).

(5)

5. order statistics with the standard exponential : Z1, . . . , Zn IID Z ⇠ exp (1), n 2 N

a) With Z0 =0, determine the joint distribution of the coverages, Z(i)�Z(i�1), i = 1, . . . , n.

(12)

b) Using the result in a), or otherwise, obtain EZ(i) and �(Z(i)) for every 1  i  n as well as the correlation coe�cients ⇢(Z(i), Z(j)) for 1  i < j  n

(8)

REQUIRED: AGREEMENT TO ACADEMIC INTEGRITY 2

All suspected cases of academic dishonesty will be investigated following the procedures outlined

in the Code of Behaviour on Academic Matters.

In order to complete this term test, you are here required to sign the Statement below.

By signing this Statement, I , (print your full name here),

(student number here),

am attesting to the fact that I have abided fully to the Code of Behaviour on Academic Mat-

ters. I have not committed academic misconduct, and am aware of the penalties that may be

imposed if I have committed an academic o↵ence.

signed

Thank you.