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Chapter 9
Random
Variables
Copyright © 2014, 2011 Pearson Education, Inc.
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9.1 Random Variables
Will the price of a stock go up or down?

Need language to describe processes that show
random behavior (such as stock returns)

“Random variables” are the main components of
this language
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9.1 Random Variables
Definition of a Random Variable

Describes the uncertain outcomes of a
random process

Denoted by X

Defined by listing all possible outcomes
and their associated probabilities
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9.1 Random Variables
Suppose a day trader buys one share of IBM

Let X represent the change in price of IBM

She pays $100 today, and the price
tomorrow can be either $105, $100 or $95
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9.1 Random Variables
How X is Defined
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9.1 Random Variables
Two Types: Discrete vs. Continuous


Discrete – A random variable that takes on
one of a list of possible values (counts)
Continuous – A random variable that takes
on any value in an interval
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9.1 Random Variables
Graphs of Random Variables

Show the probability distribution for a
random variable

Show probabilities, not relative frequencies
from data
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9.1 Random Variables
Graph of X = Change in Price of IBM
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9.1 Random Variables
Random Variables as Models

A random variable is a statistical model

A random variable represents a simplified
or idealized view of reality

Data affect the choice of probability
distribution for a random variable
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9.2 Properties of Random Variables
Parameters

Characteristics of a random variable, such
as its mean or standard deviation

Denoted typically by Greek letters
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9.2 Properties of Random Variables
Mean (µ) of a Random Variable

Weighted sum of possible values with
probabilities as weights
  x1 px1   x2 px2   ...  xk pxk 
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9.2 Properties of Random Variables
Mean (µ) of X (Change in Price of IBM)
  5 p 5  0 p0  5 p5
 50.09  00.80  50.11
 $.10
The day trader expects to make 10 cents on
every share (on average) of IBM she buys.
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9.2 Properties of Random Variables
Mean (µ) as the Balancing Point
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9.2 Properties of Random Variables
Mean (µ) of a Random Variable

Is a special case of the more general
concept of an expected value, E(X)
E  X     x1 px1   x2 px2   ...  xk pxk 
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9.2 Properties of Random Variables
Caution – Expected Value
The expected value of a random variable
may not match one of the possible outcomes
as it represents a long run average. As in the
IBM stock example, the price never changes
by 10 cents.
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9.2 Properties of Random Variables
Variance (σ2) and Standard Deviation (σ)

The variance of X is the expected value of
the squared deviation from µ
 2  Var  X 
 EX   
2
  x1    p x1    x2    p x2   ...   xk    p xk 
2
2
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9.2 Properties of Random Variables
Calculating the Variance (σ2 ) for X
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9.2 Properties of Random Variables
Calculating the Variance (σ2 ) for X
 2  Var  X 
  5  0.10 0.09  0  0.10 0.80  5  0.10 0.11
2
2
2
 4.99
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9.2 Properties of Random Variables
The Standard Deviation (σ ) for X
  SD X   Var  X 
 4.99  $2.23
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Motivation
CheapO Computers shipped two servers to its biggest
client. Four refurbished computers were mistakenly
restocked along with 11 new systems. If the client
receives two new servers, the profit for the company is
$10,000; if the client receives one new server, the profit is
$4,500. If the client receives two refurbished systems, the
company loses $1000. What are the expected value and
standard deviation of CheapO’s profits?
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Method
Identify the relevant random variable, X,
which is the amount of profit earned on this
order. Determine the associated probabilities
for its values using a tree diagram. Compute
µ and σ.
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Mechanics – Tree Diagram
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Mechanics – Probabilities for X
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Mechanics – Compute µ and σ
E(X) = µ = $7,067
Var(X) = σ2 = 10,986,032 $2
SD(X) = σ = $3,315
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4M Example 9.1:
COMPUTER SHIPMENTS & QUALITY
Message
This is a very profitable deal on average. The large
standard deviation is a reminder that profits are
wiped out if the client receives two refurbished
systems.
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9.3 Properties of Expected Values
Adding or Subtracting a Constant (c)

Changes the expected value by a fixed
amount:
E(X ± c) = E(X) ± c

Does not change the variance or standard
deviation:
Var(X ± c) = Var(X)
SD(X ± c) = SD(X)
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9.3 Properties of Expected Values
Subtracting c from Expected Value
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9.3 Properties of Expected Values
Multiplying by a Constant (c)

Changes the mean and standard deviation
by a factor of c: E(cX) = c E(X)
SD(cX) = |c| SD(X)

Changes the variance by a factor of c2:
Var(cX) = c2 Var(X)
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9.3 Properties of Expected Values
Multiplying Expected Value by c
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9.3 Properties of Expected Values
Rules for Expected Values (a and b are
constants)

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
E(a + bX) = a + bE(X)
SD(a + b X) = |b|SD(X)
Var(a + bX) = b2Var(X)
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9.4 Comparing Random Variables

May require transforming random variables
into new ones that have a common scale

May require adjusting if the results from the
mean and standard deviation are mixed
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9.4 Comparing Random Variables
The Sharpe Ratio


Popular in finance
Is the ratio of an investment’s net expected
gain to its standard deviation
  rf
S X  

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9.4 Comparing Random Variables
The Sharpe Ratio – An Example
S(Apple) = 0.151
S(McDonald’s) = 0.135
Apple is preferred to McDonald’s
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Best Practices

Use random variables to represent uncertain
outcomes.

Draw the random variable.

Recognize that random variables represent
models.

Keep track of the units of a random variable.
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Pitfalls

Do not confuse x with µ or s with σ.

Do not mix up X with x.

Do not forget to square constants in variances.
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