Statistics for business decision making and analysis robert stine and foster chapter 09

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Random Variables

Chapter 9

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9.1 Random Variables

Will the price of a stock go up or down?

random behavior (such as stock returns)

this language

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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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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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How X is Defined

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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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Graphs of Random Variables

 Show the probability distribution for a

random variable

 Show probabilities, not relative frequencies from data

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Graph of X = Change in Price of IBM

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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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Mean (µ) of a Random Variable

 Weighted sum of possible values with

probabilities as weights

  x x p   x xk p   xkp



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Mean (µ) of X (Change in Price of IBM)

The day trader expects on average to make

10 cents on every share of IBM she buys

0 5 80

0 0 09

0 5

5 5

0 0

5 5

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Mean (µ) as the Balancing Point

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Mean (µ) of a Random Variable

 Is a special case of the more general

concept of an expected value, E(X)

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 X x p x x p x x k p x k

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Variance (σ2) and Standard Deviation (σ)

 The variance of X is the expected value of

the squared deviation from µ

 

x  p  x x  p x x k  p x k

X E

X Var

2 2

1

2 1

2 2

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Calculating the Variance (σ2 ) for X

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Calculating the Variance (σ2 ) for X

 

99

4

11 0 10

0 5

80 0 10

0 0

09 0 10

0

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The Standard Deviation (σ ) for X

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23 2

$ 99

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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 among 11 new systems If the client receives

two new systems, the profit for the company is $10,000; if the client receives one new system, the profit is $9,600 If the client receives two refurbished systems, the company loses $800 What are the expected value and standard

deviation of CheapO’s profits?

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4M Example 9.1:

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:

Mechanics – Tree Diagram

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4M Example 9.1:

Mechanics – Probabilities for X

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4M Example 9.1:

Mechanics – Compute µ and σ

E(X) = µ = $9,215

Var(X) = σ2 = 6,116,340 $2

SD(X) = σ = $2,473

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4M Example 9.1:

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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Multiplying by a Constant (c)

 Changes the mean and standard deviation

by a factor of c: E(cX) = c E(X)

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Rules for Expected Values (a and b are

constants)

 E(a + bX) = a + bE(X)

 SD(a + b X) = |b|SD(X)

 Var(a + bX) = b 2 Var(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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The Sharpe Ratio

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The Sharpe Ratio – An Example

S(Disney) = 0.0253

S(McDonald’s) = 0.0171

Disney is preferred to McDonald’s

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Best Practices

outcomes.

models.

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x

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