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University of British Columbia Probability Exam - Math 418, Term 1, 2006, Exams of Probability and Statistics

AMET UniversityProbability and Statistics

This is a 3-page final exam for the university of british columbia's math 418 probability course offered in term 1, 2006. The exam consists of 7 questions and covers topics such as random walks, conditional expectation, uniform distributions, distance calculations, branching processes, and the central limit theorem. A table of the normal distribution is provided.

Typology: Exams

2012/2013

Uploaded on 02/25/2013

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Be sure this exam has 3 pages including the cover
The University of British Columbia
Sessional Exams – 2006 Term 1
Mathematics 418 Probability
Dr. G. Slade
This exam consists of 7questions worth 10 marks each.
No calculators or other aids are permitted.
Show all work and calculations and explain your reasoning thoroughly.
A table of the normal distribution is on the last page.
1. Each candidate should be prepared to produce his library/AMS card upon request.
2. Read and observe the following rules:
No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave
during the first half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness
shall not be received.
3. Smoking is not permitted during examinations.
pf3

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Download University of British Columbia Probability Exam - Math 418, Term 1, 2006 and more Exams Probability and Statistics in PDF only on Docsity!

Be sure this exam has 3 pages including the cover

The University of British Columbia

Sessional Exams – 2006 Term 1

Mathematics 418 Probability

Dr. G. Slade

This exam consists of 7 questions worth 10 marks each.

No calculators or other aids are permitted.

Show all work and calculations and explain your reasoning thoroughly.

A table of the normal distribution is on the last page.

  1. Each candidate should be prepared to produce his library/AMS card upon request.
  2. Read and observe the following rules:

No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave

during the first half hour of the examination.

Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities

in examination questions.

CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the

examination and shall be liable to disciplinary action.

(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.

(b) Speaking or communicating with other candidates.

(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness

shall not be received.

  1. Smoking is not permitted during examinations.

December 19 2006 Math 418 Final Exam Page 2 of 3

1. Prove the identity

∑n

k=

n + k

k

2 −(n+k)^ = 1.

Hint: Consider a random walk on the square lattice Z^2 which starts at the origin (0, 0) and

takes independent steps either upward (from (x, y) to (x, y + 1)) or the the right (from (x, y)

to (x + 1, y)) with equal probabilities

2. Note that it eventually leaves the square [0, n]^ ×^ [0, n].

2. Let X and Y be discrete random variables.

(a) Prove directly from the definition of conditional expectation that

E(XY ) = E[XE(Y |X)].

(b) Prove that if

E(Y |X = x) = E(Y ) for all x,

then X and Y are uncorrelated.

(c) Give a counterexample to show that the converse of (b) is false.

3. Let X be uniform on [0, 1] and Y be uniform on [0, X 2 ].

(a) Find the marginal density of Y.

(b) Find E(X|Y ).

4. Suppose n darts are thrown at a circular target of radius 1. Assume the darts land at

positions which are i.i.d. random variables uniform on the unit disk. Let Zn be the distance

from the centre of the target to the closest dart. Show that E(Zn) =

0 (1^ −^ x

2 )ndx.

5. Consider a branching process whose family sizes have the geometric mass function

f (k) = 2−k−^1 , for k ≥ 0. Let Zn be the size of the nth generation, and assume that Z 0 = 1.

(a) Show that Zn has generating function

Gn(s) =

n − (n − 1)s

n + 1 − ns

(b) Let T = min{n : Zn = 0} be the extinction time. Find P (T = n) for all n ≥ 1.

(c) Prove that P (T < ∞) = 1.

6. Let λ > 0. For n = 1, 2 , 3 ,.. ., let Xn have a Binomial(n, λ/n) distribution. Let X have a

Poisson(λ) distribution. Prove that Xn converges to X in distribution.

7. An astronomer is interested in measuring, in light years, the distance from her observatory to

a distant star. Although she has a measuring technique, she knows that because of changing

atmospheric conditions and experimental error, each time a measurement is made it will not

yield the exact value but rather an approximate value. As a result the astronomer plans to

make a series of measurements and then use the average of these measurements as the

estimated value of the actual distance. If the astronomer believes that the values of the

measurement errors are not systematic, so that the measurements are described by a random

variable with mean d (the true distance) and a variance of 4 light years, use the central limit

theorem to determine approximately the number of measurements that should be made to be

95% sure that the estimated distance is accurate to within ± 0 .5 light years.