Module Code: MTHE03C02

Final Examination; 2011/2012

Module Title Probability and Statistics Module Leader Dr. Khaled Salem

Semester One

Equipment allowed: Scientific calculator

Instructions to Students

• Answer only four questions of section A and all questions of section B. • The exam paper is four pages long excluding this page, two pages of

questions, a page of useful formulas, and a table. • The allocation of marks is shown in brackets by the questions.

This examination is 2 hours long (2 hrs).

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MTHE03C02 Probability and Statistics

Section A (Answer only four questions)

Q A1. A small software company employs four programmers. The percentage ofcode written by each programmer and the percentage of errors in theircode are shown in the following table.

Percentage of Code Percentage of Errors

Programmer 1 15 5Programmer 2 20 3Programmer 3 25 2Programmer 4 40 1

If a code, selected at random, is examined and found to be free of errors,compute the probabilities that it was written by

i. programmer 1,

ii. programmer 2,

iii. programmer 3,

iv. programmer 4.

[Total 15 marks]

Q A2. Suppose A and B are events with 0 < P (A) < 1 and 0 < P (B) < 1.Prove that

i. If A and B are disjoint, then A and B cannot be independent.

ii. If A ⊆ B, then A and B cannot be independent.

iii. If A and B are independent, then A and A∪B cannot be independent.

[Total 15 marks]

Q A3. Five men and five women are ranked according to their scores on anexamination. Assume that no two scores are alike and all 10! possiblerankings are equally likely. Let X denotes the highest ranking achievedby a woman (for instance, X = 2 if the top-ranked person was male andthe next-ranked person was female). Find the probability mass functionof X.

[Total 15 marks]

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MTHE03C02 Probability and Statistics

Q A4. An insurance company writes a policy to the effect that an amount ofmoney A must be paid if some event E occurs within a year. If thecompany estimates that E will occur within a year with probability p,compute the amount of money it should charge the customer so that itsexpected profit will be 10% of A.

[Total 15 marks]

Q A5. The density function of X is given by

f(x) =

{a+ bx2 0 ≤ x ≤ 10 otherwise

If E(X) = 35 , find a and b.

[Total 15 marks]

Section B (Answer all questions)

Q B1. Jones figures that the total number of thousands of miles that a car canbe driven before it would need to be scrapped is an exponential randomvariable with parameter 1

20 . Smith has a used car that he claims he hasdriven only 10 thousands miles. If Jones purchases the car, what is theprobability that she would get at least 20 thousands additional milesout of it? Repeat under the assumption that the lifetime mileage of thecar is not exponentially distributed but rather is (in thousands of miles)uniformly distributed over [0, 40].

[Total 20 marks]

Q B2. If X is a continuous random variable having distribution function F , thenits median is defined as that value of m for which

F (m) =1

2Find the median of the random variables:

i. an exponential random variable with parameter λ = 1.

ii. a uniform random variable over the interval [0, 1].

[Total 20 marks]

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MTHE03C02 Probability and Statistics

Useful formulaeSpecial random variables

The Binomial Random variable

P (X = x) =

(n

x

)px(1− p)n−x, x = 0, 1, 2, . . . , n

E[X] = np, V ar(X) = np(1− p)The Geometric Random Variable

P (X = x) = (1− p)x−1p, x = 1, 2, 3, . . .

E[X] =1

pV ar(X) =

1− pp2

The Poisson Random Variable

P (X = x) = e−λλx

x!, x = 0, 1, 2, . . .

E[X] = V ar(X) = λ

The Uniform Random Variable

F (x) =

0, x < αx−αβ−α , α ≤ x ≤ β

1, β < x

E[X] =β + α

2, V ar(X) =

(β − α)2

12Normal Random Variables

F (x) = Φ

(x− µσ

).

E[X] = µ, V ar(X) = σ2

Exponential Random Variables

F (x) =

{1− e−λx x ≥ 00 x < 0

E[X] =1

λ, V ar(X) =

1

λ2

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