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Copyright © 2014, 2011 Pearson Education, Inc 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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Copyright © 2014, 2011 Pearson Education, Inc. 1
Chapter 9Random Variables
Copyright © 2014, 2011 Pearson Education, Inc. 2
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
Copyright © 2014, 2011 Pearson Education, Inc. 3
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
Copyright © 2014, 2011 Pearson Education, Inc. 4
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 $95
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9.1 Random Variables
How X is Defined
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9.1 Random Variables
Two Types: Discrete vs. Continuous
Discrete – A random variable that takes on one of a list of possible values (counts)
Continuous – A random variable that takes on any value in an interval
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9.1 Random Variables
Graphs of Random Variables
Show the probability distribution for a random variable
Show probabilities, not relative frequencies from data
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9.1 Random Variables
Graph of X = Change in Price of IBM
Copyright © 2014, 2011 Pearson Education, Inc. 9
9.1 Random Variables
Random Variables as Models
A random variable is a statistical model
A random variable represents a simplified or idealized view of reality
Data affect the choice of probability distribution for a random variable
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9.2 Properties of Random Variables
Parameters
Characteristics of a random variable, such as its mean or standard deviation
Denoted typically by Greek letters
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9.2 Properties of Random Variables
Mean (µ) of a Random Variable
Weighted sum of possible values with probabilities as weights
kk xpxxpxxpx ...2211
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9.2 Properties of Random Variables
Mean (µ) of X (Change in Price of IBM)
The day trader expects to make 10 cents on every share (on average) of IBM she buys.
10$.
11.0580.0009.05550055
ppp
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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)
kk xpxxpxxpxXE ...2211
Copyright © 2014, 2011 Pearson Education, Inc. 15
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
kk xpxxpxxpx
XE
XVar
22
221
21
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
99.4
11.010.0580.010.0009.010.05 222
2
XVar
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9.2 Properties of Random Variables
The Standard Deviation (σ ) for X
23.2$99.4
XVarXSD
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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?
Copyright © 2014, 2011 Pearson Education, Inc. 21
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)
Copyright © 2014, 2011 Pearson Education, Inc. 29
9.3 Properties of Expected ValuesMultiplying Expected Value by c
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9.3 Properties of Expected Values
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) = 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
frXS
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9.4 Comparing Random Variables
The Sharpe Ratio – An Example
S(Apple) = 0.151S(McDonald’s) = 0.135Apple 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 with µ or s with σ.
Do not mix up X with x.
Do not forget to square constants in variances.
x