Copyright © 2011 Pearson Education, Inc. Random Variables Chapter 9.

transcript

Random Variables

Chapter 9

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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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

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kk xpxxpxxpx ...2211

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.

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11.0580.0009.05

550055

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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kk xpxxpxxpxXE ...2211

Variance (σ2) and Standard Deviation (σ)

The variance of X is the expected value of the squared deviation from µ

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kk xpxxpxxpx

Calculating the Variance (σ2 ) for X

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Calculating the Variance (σ2 ) for X

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11.010.0580.010.0009.010.05 222

The Standard Deviation (σ ) for X

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23.2$99.4

XVarXSD

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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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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Mechanics – Tree Diagram

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Mechanics – Probabilities for X

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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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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)

SD(cX) = |c| SD(X)

Changes the variance by a factor of c2:Var(cX) = c2 Var(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) = 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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The Sharpe Ratio

Popular in finance Is the ratio of an investment’s net expected

gain to its standard deviation

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The Sharpe Ratio – An Example

S(Disney) = 0.0253S(McDonald’s) = 0.0171Disney 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.

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Copyright © 2011 Pearson Education, Inc. Random Variables Chapter 9.

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