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University of British Columbia Math 418 Probability Exam - December 2011, Exams of Probability and Statistics

AMET UniversityProbability and Statistics

A final exam for mathematics 418 probability at the university of british columbia, held in december 2011. The exam consists of 7 questions worth 10 marks each, and no calculators or other aids are permitted. The questions cover topics such as binomial distributions, geometric distributions, normal distributions, and expectations.

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 – 2011 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 pro duce 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 p ermitted during examinations.
pf3

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Download University of British Columbia Math 418 Probability Exam - December 2011 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 – 2011 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 6, 2011 Math 418 Final Exam Page 2 of 3

1. Suppose that a nonempty subset is chosen uniformly at random from a set of n elements, in

the sense that every nonempty subset is equally likely to be selected. Let X be the number of

elements in the chosen subset. Hint: recall that the mean and variance of a Bin(n, p) random

variable are respectively np and np(1 − p).

(a) Show that

EX =

n

2 − 21 −n^

(b) Show that

VarX =

n 22 n−^2 − n(n + 1)2n−^2

(2n^ − 1)^2

2. Suppose that X 1 and X 2 are independent binomial random variables with parameters (n 1 , p)

and (n 2 , p), respectively. Prove that X 1 + X 2 is a binomial random variable with parameters

(n 1 + n 2 , p).

3. A geometric random variable with parameter p ∈ (0, 1) has probability mass function

P (X = n) = (1 − p)n−^1 p, n = 1, 2 , 3 ,.. .. Let X 1 and X 2 be independent geometric random

variables, both with parameter p. Find:

(a) P (X 1 = X 2 ),

(b) P (X 1 ≥ X 2 ).

4. Let Xn be independent N (0, 1) random variables. Prove that

P

lim sup

n→∞

|Xn|

2 log n

Hint: Recall that the cumulative distribution function Φ of the standard normal distribution

obeys:

(x

− x

)e

−x^2 / 2

2 π[1 − Φ(x)] < x

e

−x^2 / 2

for x > 0.

5. Suppose f : R → R is bounded and is continuous at 0. Suppose also that Xn are random

variables such that Xn → 0 in probability (we are not assuming almost sure convergence).

Prove that limn→∞ E(f (Xn)) = f (0).

6. A certain immigration clerk will answer the telephone only on weekdays at about 10:00 a.m.

On any such morning she is equally likely to be at her desk or not. If she is absent no one

answers, and days are independent. Assume the line is never busy. If she is at her desk, the

time T that she takes to answer her phone is a random variable with distribution function

given by FT (t) = P (T ≤ t) = 0 for t ≤ 1 and FT (t) = P (T ≤ t) = 1 − t−^1 for t > 1.

(a) If you telephone this clerk one morning, and do not hang up, what is the probability that

the phone rings for a time R that is at least time s?

(b) You decide to telephone the clerk one morning, and to hang up at time s if she has not

answered by then. If your call is successful, then the expected time for which the

telephone rings is E(R|R < s). Show that

E(R|R < s) =

s log s

s − 1

, for s > 1.

Hint: the expectation of a nonnegative continuous random variable X is

0 P^ (X > t)dt.