Download Mathematics for Economics: Exercise 1

QM1 - MODULE 11122
EXERCISE 1 - RANDOM VARIABLES AND THEIR DISTRIBUTIONS
Although you should attempt all questions as the material is covered in the lectures, only questions marked
with a T will be discussed in tutorials which start in week 2, so it is important that you especially attempt
these before your tutorials. Only questions marked with an EL will be covered in the Exercises lectures (
Alternate Mondays at 2 p.m.) and it is also important that you try these before the Exercise lectures. Solutions
to those not covered in tutorials or Exercise lectures will be provided later on in the module.
1.
An investment consultant on the basis of his past experience has suggested the following
probability model for the rate of return of the investment.
Rate of return (%)
10
11
12
13
14
15
16
Probability
0.05
0.10
0.15
0.17
0.12
0.08
0.09
Rate of return (%)
17
18
19
20
21
Probability
0.06
0.05
0.05
0.04
0.04
(i) Define the random variable here.
(ii) Calculate the probability of a rate of return of less than 14%.
(iii) Calculate the probability of a rate of return of at least 14%
(iv) Calculate the expected return on this investment and the investment risk.
2.
The managing director of North Star Stoves is considering a project to increase production.
The possible returns from the project are: £3000, £9000, £15000, and £21000 with probabilities
0.30, 0.20, 0.40 and 0.10 respectively.
(i)
Define the investment probability model.
(ii)
Find the probability that the return is (a) at least £15000 (b) between £9000 and £21000 inclusive.
(iii) The firm’s banker has asked for a single value for the return on investment. Select the
appropriate summary measure and compute its value.
(iv) The managing director is interested in the variability of returns for the proposed
investment . Calculate the standard deviation of returns.
T3.
The manager of Quality Computers Ltd. has asked for your assistance in analysing daily computer sales
of
a particular computer at its retail stores. The sales pattern for this computer at each store is similar. The
following table shows the probability distribution of sales of this computer.
X
0
1
p(x)
0.10 0.10
2
3
0.10 0.20
4
5
0.20 0.10
6
0.10
7
8
0.05
0.05
(i) Construct the cumulative probability distribution function.
(ii) What is the probability of selling at least two computers but fewer than seven?
(iii) The stock of computers at the beginning of a particular day is 5 computers. What is the probability
that sales exceed the available stock.
(iv) The inventory at the beginning of the day is 15 computers. What is the probability that the inventory
at the end of the day is 10 or more?
(v) What is the mean number of computers sold each day?.
2
(vi)
What is the standard deviation of computers sold each day?.
The president of the company wishes to study the sales revenue pattern for the company stores and
the daily pattern of contribution to profit and overhead. Each computer sells for £1200.
Contribution is the total revenue minus the cost of the goods sold minus the daily fixed costs of £500.
Each computer of this type costs the company £900.
(vii) Find the mean daily sales revenue of this computer and the variance of the daily sales revenue.
(viii) Find the mean contribution to profit and overhead and the variance of the contribution.
(ix) Find the probability that the contribution is greater than zero.
4.
A continuous random variable X has a distribution with p.d.f. given by
3
(4  x 2 )
16
=0
f ( x) 
0x2
otherwise
(i) Sketch the p.d.f. curve.
(ii) Find the following probabilities (a) P ( X > 1)
(iii) Show that E( X ) = 43 and Var( X ) = 19
80 .
5.
(b) P ( 0.5
X
1.7)
The probability density function of a random variable X, f (x) is given below,
f  x  0
 ax
x0
0 x
 a1  x 
1
2
0
1 x
1
2
 x 1
(i)
Sketch the p.d.f. function.
(ii)
What value must ‘a’ take if is to be a proper probability density function.
(iii)
Calculate the following probabilities (a) P ( X > 0.25 ), (b) P ( 0.75 < X < 1.3 ) .
(iv)
Find E( X) and stdev (X).
(v)
Find the cumulative distribution function, F(x), corresponding to f (x).
[ Hint for part (iii) and E(X) : you can use calculus or geometrical arguments so look at sketch.]
[ Hint for part (v): the c.d.f. will have 4 parts to it as with f (x) ] .
EL6 .
An economic random variable X has a p.d.f. given by
015
. x2
0 x2

f  x   0.30 4  x 
2x4
0
otherwise

(i)
(ii)
(iii)
(iv)
(v)
Graph the p.d.f
Find the mean of X.
Find the lower quartile, the upper quartile and the median value of X.
Find the probability that a single observation X lies between the lower quartile and the mean.
Three independent observations of X are taken. Find the probability that one of the observations is
greater than the mean and the other two are less than the median value.
3
EL7.
A common model for the distribution of incomes is one based on the Pareto distribution
which has a distribution function given by F(x) where
 A
F  x  1   
x
(i)

if x
A
and F(x) = 0 if x < A where A,  > 0
Sketch F(x) .
(ii) Derive the density function of X and sketch it. (Note that F(x) is zero for x < A).
    r  . A
(iii) Show that E X r 
r
provided r <  , and hence find an expression for
the mean and variance of the Pareto distribution.
(iv) Given data on a particular distribution of incomes, how could you tell whether the
Pareto distribution provided an adequate model for your income data?
C. Osborne October 2001