Chapter 4
Chapter 4
Tim Low
tim.low@uct.ac.za
Room 5.65
University of Cape Town
Wednesday 12 th August 2015
Chapter 4
Random variables
Some Definitions
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A sample space is a set of all the elementary events that are
possible outcomes of a random experiment
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Elementary events may be expressed quantitatively or
qualitatively
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To manipulate events defined on a sample space
mathematically, it is necessary to assign a numerical value to
each elementary event
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For example, consider a random experiment in which a coin is
tossed. We could assign a “0” to the event that a head is
observed and a “1” to the event that a tail is observed
Chapter 4
Random variables
Definition of a random variable
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Once numerical values have been assigned, let X “stand for”
the numerical values of the elementary events
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X is then called a random variable because it can take on
different values depending on the outcome of a random
experiment
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Random variables are denoted by capital letters
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X is a variable because it can take on the various different
values in the sample space
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X is random because the particular value it takes on depends
on the outcome of a random experiment which is not known
in advance
Chapter 4
Random variables
Definition of a random variable
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When dealing with random variables, events are defined as
subsets of the real line.
Eg. of events: X = 1 ; 5 < X < 10
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We can assign probabilities to these events
e.g. P[X = 1] ; Pr[5 < X < 10]
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The particular values that a random variable may assume are
denoted by small letters.
Eg. P[X = x] is the probability that the random variable X
takes on the value x. If there is only 1 random variable this
can be abbreviated as P(x).
For rolling a die: P(x) = 1/6 for x=1, 2, 3, 4, 5 and 6.
Chapter 4
Random variables
Example
Example 5C
A car salesperson is scheduled to see two clients today. She sells
only two models of cars, an executive (E) and basic (B) model.
Each executive model sold earns the salesperson a commission of
R2000, while each basic model sold earns her only R1000. If the
sale is lost (L), no commission is earned.
Suppose P(E)=0.2, P(B)=0.3 and P(L)=0.5, and that sales are
independent of each other. Let the r.v. X be the total commission
earned today. What values can X take on, and with what
probabilities?
Chapter 4
Random variables
Discrete and Continuous Random Variables
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Discrete random variables
Take on isolated values along the real line
Eg: The number of spectators at a soccer match
The number of occupied tables at a restaurant
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Continuous random variables:
Can be measured to any degree of accuracy
Eg. The distance a car travels on one litre of petrol
The time that a customer waits in the queue at Steers
Chapter 4
Random variables
Discrete and Continuous Random Variables
Example 6C
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Which of the following are random variables? Which of the random
variables are discrete and which are continuous? Write down the set
of values that each random variable can take on.
a The number of customers arriving at a supermarket during the
morning
b The number of letters in the Greek alphabet
c The opening price of gold in New York on Monday next week
d The number of seats that will be sold for a performance of a
play in a theatre with a capacity of 328 seats
e The length of time you have to wait at an autobank
f The ratio between the diameter and circumference of a circle
g The last digit of a randomly selected telephone number
Chapter 4
Random variables
Probability Mass Functions (PMF)
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Discrete random variables are represented by PMFs
A function p(x) is a PMF if it satisfies the following
conditions:
1. p(x) is defined for all values of x but p(x) 6= 0 at a finite set of
points
2. P
0 ≤ p(x) ≤ 1
3.
p(x) = 1
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Probabilities for discrete random variables are found by
calculating the values of the PMF p(x) at the points of
interest and summing them:
P[a ≤ X ≤ b] =
b
X
x=a
p(x) or P[a < X < b] =
b−1
X
x=a+1
p(x)
Chapter 4
Random variables
Examples
Example 11A
(a) Check that the following function satisfies the conditions for a
PMF:
x
p(x) = 15
x = 1, 2, 3, 4, 5
= 0 Otherwise
(b) Find P[2 ≤ X ≤ 4]
(c) Sketch p(x)
Chapter 4
Random variables
Examples
Eg. 9C The Minister of Environment Affairs has to decide on a fishing
quota for the forthcoming season. Currently, the biomass of
fish is estimated to be 20 m tonnes. The fish may have a
good breeding season (with probability 0.3) and produce 10 m
tonnes of young, or have a bad breeding season and produce
only 1 m tonnes. A so called “warm-water event” may occur
with probability 0.1, and kill 15 m tonnes of fish, otherwise 1
m tonnes of fish will die. Find the probability mass function
for X, the biomass of fish before setting the quota (assuming
all events are independent). If the minister bases his decision
using a policy that the biomass must remain 10 m tonnes or
more with probability 0.8, what should his decision be?
Chapter 4
Random variables
Examples
Eg. 10C The hostile merger bid by Minorco on Consgold in 1989 was,
at one point, considered highly likely to fail by the financial
media. They quoted a 12-1 chance of failure. Express the
anticipated outcome of the merger as a probability mass
function
Eg. 2A Consider a random experiment that consists of tossing an
unbiased coin three times. Write out the elements in S. Find
the PMF of the random variable X, the number of heads
observed in the random experiment
Chapter 4
Random variables
Examples
Eg. 13C
Find the sample space for the random experiment which consists of
rolling a pair of dice. Find the probability mass function for the
random variable X defined to be the sum of the values on the dice
and Y defined to be the product of the values. Find P[X ≥ 10]
and P[Y ≥ 13].
The outcomes of tossing the 2 die are:

(1, 1) (1, 2) (1, 3) (1, 4)
 (2, 1) (2, 2) (2, 3) (2, 4)

 (3, 1) (3, 2) (3, 3) (3, 4)

 (4, 1) (4, 2) (4, 3) (4, 4)

 (5, 1) (5, 2) (5, 3) (5, 4)
(6, 1) (6, 2) (6, 3) (6, 4)
(1, 5)
(2, 5)
(3, 5)
(4, 5)
(5, 5)
(6, 5)
(1, 6)
(2, 6)
(3, 6)
(4, 6)
(5, 6)
(6, 6)








Chapter 4
Random variables
Probability Density Functions (PDF)
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Continuous random variables are represented by PDFs
A function f (x) is a PDF if it satisfies the following
conditions:
1. f (x) is defined for all values of x
2. 0
≤ f (x) ≤ ∞
R∞
3. −∞ f (x)dx = 1 i.e. the “area under the curve” for a pdf will
equal 1.
Chapter 4
Random variables
Probability Density Functions (PDF)
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For continuous random variables, the PDF f (x) is constructed
in such a way that probabilities of events are found by
integration
The probability that X falls between a and b is found by
calculating the area under the graph of f (x) between a and b.
Chapter 4
Random variables
Probability Density Functions (PDF)
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NB: For discrete random variables, the PMF p(x) is the
probability that the r.v. X takes on the value x. Note,
however, that for continuous r.v.s, the PDF f(x) is NOT the
probability that X takes on x.
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The probability that a continuous random variable will assume
any particular value exactly is zero.
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It is only meaningful to talk about X assuming a value within
a particular interval.
P[X = a] = Pr [X = b] = 0
P[a ≤ X ≤ b] = P[a < X ≤ b] = P[a ≤ X < b] = P[a < X < b]
Chapter 4
Random variables
Examples
Eg. 21B In a certain risky sector of the share market, the proportion of
companies that survive (i.e. are not delisted) a year is a
continuous random variable lying in the interval from zero to
one. A statistician examines the data collected over past years
and suggests that the function
f (x) = 20x 3 (1 − x) 0 ≤ x ≤ 1
=
0
Otherwise
might be useful in modeling X, the annual proportion of
companies that survive.
(a) Check that f (x) is a probability density function.
(b) What is the probability that between 30% and 50% of the
companies survive a year?
(c) What is the probability that less than 10% of the companies
survive a year?
Chapter 4
Random variables
Examples
Eg. 25C (a) Find the value of k so that the function
f (x)
=
=
k(x 2 − 1) 1 ≤ x ≤ 3
0
Otherwise
may serve as a pdf.
(b) Find the probability that X lies between 2 and 3
Chapter 4
Random variables
Examples
Eg. 27C The probability density function of a random variable X is
given by:
f (x) = kx(1 − x 2 ) 0 ≤ x ≤ 1
=
0
Otherwise
(a) Show that the value of k must be 4
(b) Calculate Pr [0 < X < 21 ]
Chapter 4
Random variables
Examples
A continuous
random variable X has probability density function
0.25 for 1 ≤ x ≤ b
f (x) =
0
otherwise
(a) Find the value of b
(b) Find the probability that X > 2
Chapter 4
Random variables
Examples
A random
 variable
 4x
f (x) =
4(1 − x)

0
X can be described by the function defined as follows:
for 0 < x < 12
for 12 ≤ x < 1
otherwise
(a) Draw a graph of the function f(x).
(b) Show that f(x) is a p.d.f for the random variable X.
(c) Find P[ 14 < X < 34 ]
Chapter 4
Random variables
Examples
Eg. 29C A small pool building company is equally likely to be able to
complete 2 or 3 pool contracts each month. The company
receives between 1 and 4 contracts to build pools each month,
with probabilities P(1) = 0.1, P(2) = 0.2, P(3) = 0.5, P(4) =
0.2. At the beginning of this month the company has two
contracts carried forward from the previous month. The
random variable X of interest is the number of contracts to be
carried forward to next month. Find the probability mass
function of X. In particular, what is the probability that no
contracts will be carried forward to next month? Assume that
the number of contracts is independent of the number of
pools completed. Also, to simplify the problem, assume that
the contracts for a month are made at the beginning of the
month.