Math 141Lecture 5: Expected Value

Albyn Jones1

1Library 304jones@reed.edu

www.people.reed.edu/∼jones/courses/141

Albyn Jones Math 141

History

The early history of probability theory is intimately related toquestions arising in gambling.

For example:

Two gamblers playing a game that ends when one has wonall of the other’s money have to quit in the middle of thegame. What is the fair division of the pot?What is the fair price of a lottery ticket?

Albyn Jones Math 141

History

The early history of probability theory is intimately related toquestions arising in gambling.

For example:Two gamblers playing a game that ends when one has wonall of the other’s money have to quit in the middle of thegame. What is the fair division of the pot?

What is the fair price of a lottery ticket?

Albyn Jones Math 141

History

The early history of probability theory is intimately related toquestions arising in gambling.

For example:Two gamblers playing a game that ends when one has wonall of the other’s money have to quit in the middle of thegame. What is the fair division of the pot?What is the fair price of a lottery ticket?

Albyn Jones Math 141

Reading!

READ CHAPTER 4!

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Expected Value

Definition: Let X be a random variable with sample space

Ω = x0, x1, x2, . . .

with corresponding probabilities

P(X = x0) = p0, P(X = x1) = p1, P(X = x2) = p2, . . .

Then the expected value of X , denoted E(X ) is

E(X ) = x0p0 + x1p1 + x2p2 . . .

Expected value is just the weighted average of the set ofpossible outcomes, with weights equal to the probability eachoutcome occurs.

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Example: Y ∼ Bernoulli(p)

Bernoulli Trials: Each trial results in either a 1 or a 0.

P(Y = 1) = p P(Y = 0) = (1− p) = q

Expected Value:

E(Y ) = (0 · q) + (1 · p) = p

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Example: Y ∼ Bernoulli(p)

Bernoulli Trials: Each trial results in either a 1 or a 0.

P(Y = 1) = p P(Y = 0) = (1− p) = q

Expected Value:

E(Y ) = (0 · q) + (1 · p) = p

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Die Rolls: Discrete Uniform

Let X be the outcome of the roll of a fair die:

Ω = 1,2,3,4,5,6

Fair: each value occurs with probability 1/6.

E(X ) = (1 · 16

) + (2 · 16

) + (3 · 16

) + (4 · 16

) + (5 · 16

) + (6 · 16

)

Adding it up:

E(X ) =216

= 3.5

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Example: X is Geometric(p)

Probabilities: For k = 0,1,2, . . .

P(X = k) = p · qk

Expectation by Definition:

E(X ) = (0 · p · q0) + (1 · p · q1) + (2 · p · q2) + . . .

A Math 112 exercise in summing an infinite series.

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Example: X is Geometric(p)

Probabilities: For k = 0,1,2, . . .

P(X = k) = p · qk

Expectation by Definition:

E(X ) = (0 · p · q0) + (1 · p · q1) + (2 · p · q2) + . . .

A Math 112 exercise in summing an infinite series.

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Example: X is Geometric(p)

Probabilities: For k = 0,1,2, . . .

P(X = k) = p · qk

Expectation by Definition:

E(X ) = (0 · p · q0) + (1 · p · q1) + (2 · p · q2) + . . .

A Math 112 exercise in summing an infinite series.

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Expectation by Heuristic:

Consider how long we wait for each six when rolling a fairdie. In the long run, roughly 1 of every 6 rolls is a six. Thusthe long run average must be 5 non-sixes before each six.

Consider how long we wait for Heads when tossing a faircoin. In the long run, roughly 1 of every 2 tosses yieldsHeads. Thus the long run average is 1 Tail before eachHeads.

5 = 6− 1 =1

1/6− 1 =

5/61/6

1 = 2− 1 =1

1/2− 1 =

1/21/2

1p− 1 =

1− pp

=qp

Albyn Jones Math 141

Expectation by Heuristic:

Consider how long we wait for each six when rolling a fairdie. In the long run, roughly 1 of every 6 rolls is a six. Thusthe long run average must be 5 non-sixes before each six.Consider how long we wait for Heads when tossing a faircoin. In the long run, roughly 1 of every 2 tosses yieldsHeads. Thus the long run average is 1 Tail before eachHeads.

5 = 6− 1 =1

1/6− 1 =

5/61/6

1 = 2− 1 =1

1/2− 1 =

1/21/2

1p− 1 =

1− pp

=qp

Albyn Jones Math 141

Expectation by Heuristic:

Consider how long we wait for each six when rolling a fairdie. In the long run, roughly 1 of every 6 rolls is a six. Thusthe long run average must be 5 non-sixes before each six.Consider how long we wait for Heads when tossing a faircoin. In the long run, roughly 1 of every 2 tosses yieldsHeads. Thus the long run average is 1 Tail before eachHeads.

5 = 6− 1 =1

1/6− 1 =

5/61/6

1 = 2− 1 =1

1/2− 1 =

1/21/2

1p− 1 =

1− pp

=qp

Albyn Jones Math 141

Expectation by Heuristic:

Consider how long we wait for each six when rolling a fairdie. In the long run, roughly 1 of every 6 rolls is a six. Thusthe long run average must be 5 non-sixes before each six.Consider how long we wait for Heads when tossing a faircoin. In the long run, roughly 1 of every 2 tosses yieldsHeads. Thus the long run average is 1 Tail before eachHeads.

5 = 6− 1 =1

1/6− 1 =

5/61/6

1 = 2− 1 =1

1/2− 1 =

1/21/2

1p− 1 =

1− pp

=qp

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.

In the long run: Sometimes more, sometimes less, but onthe average...Gambling: The fair price of a wager.Average as typical value: The average family has 2.4children.Silly question? How many families have 2.4 children?Synonyms: average, mean, expectation value

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.In the long run: Sometimes more, sometimes less, but onthe average...

Gambling: The fair price of a wager.Average as typical value: The average family has 2.4children.Silly question? How many families have 2.4 children?Synonyms: average, mean, expectation value

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.In the long run: Sometimes more, sometimes less, but onthe average...Gambling: The fair price of a wager.

Average as typical value: The average family has 2.4children.Silly question? How many families have 2.4 children?Synonyms: average, mean, expectation value

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.In the long run: Sometimes more, sometimes less, but onthe average...Gambling: The fair price of a wager.Average as typical value: The average family has 2.4children.

Silly question? How many families have 2.4 children?Synonyms: average, mean, expectation value

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.In the long run: Sometimes more, sometimes less, but onthe average...Gambling: The fair price of a wager.Average as typical value: The average family has 2.4children.Silly question? How many families have 2.4 children?

Synonyms: average, mean, expectation value

Albyn Jones Math 141

InterpretationWhat does expected value mean?

Physics analog: Center of mass of the probabilitydistribution.In the long run: Sometimes more, sometimes less, but onthe average...Gambling: The fair price of a wager.Average as typical value: The average family has 2.4children.Silly question? How many families have 2.4 children?Synonyms: average, mean, expectation value

Albyn Jones Math 141

Median: another measure of location

Definition: MedianLet X be a RV, then any number m satisfying

P(X ≤ m) ≥ 12

andP(X ≥ m) ≥ 1

2is a median. The median of a distribution may not be unique.

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Median: examples

Consider a population with values for some variable X

1,1,2,3,4,5,10

What is the median?

3 satisfies the definition: at least half the population is lessthan or equal to 3, and at least half is greater than or equalto 3.Consider a population with values for some variable X

1,1,2,3,4,10

What is the median?Any number in the interval [2,3]!

Albyn Jones Math 141

Median: examples

Consider a population with values for some variable X

1,1,2,3,4,5,10

What is the median?3 satisfies the definition: at least half the population is lessthan or equal to 3, and at least half is greater than or equalto 3.

Consider a population with values for some variable X

1,1,2,3,4,10

What is the median?Any number in the interval [2,3]!

Albyn Jones Math 141

Median: examples

Consider a population with values for some variable X

1,1,2,3,4,5,10

What is the median?3 satisfies the definition: at least half the population is lessthan or equal to 3, and at least half is greater than or equalto 3.Consider a population with values for some variable X

1,1,2,3,4,10

What is the median?

Any number in the interval [2,3]!

Albyn Jones Math 141

Median: examples

Consider a population with values for some variable X

1,1,2,3,4,5,10

What is the median?3 satisfies the definition: at least half the population is lessthan or equal to 3, and at least half is greater than or equalto 3.Consider a population with values for some variable X

1,1,2,3,4,10

What is the median?Any number in the interval [2,3]!

Albyn Jones Math 141

Expectation as Typical Value

For a symmetric, unimodal population, the mean and medianagree, and both seem typical:

−3 −2 −1 0 1 2 3

0.00.1

0.20.3

0.40.5

Z

Dens

ity

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Expectation as Typical Value

The mean is not so typical for a multimodal or skewedpopulation:

0 5 10 15

0.00.2

0.40.6

0.8

X

Dens

ity

median

mean

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Expectation as Typical Value: Income

US Household Income: According to the US Census Bureau, in2009 the median household income in the United States was$49,777, while the mean household income $67,976.

Roughly 2/3 of all households earn less than the meanhousehold income. The US has a very skewed incomedistribution. Depending on how you define income, The top 1%of the US population gets about 24% of all income, and the top10% gets close to 50% of the total.

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Properties of Expected ValueBack to probability Theory!

Let X be a RV, and a and b constants, then

E(a + b · X ) = a + b · E(X )

Like any average, expected values are well behaved withrespect to translation (shift) and rescaling.

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Example

Let X be a Bernoulli(1/2) trial. We know E(X ) = p = 1/2. LetY = 2 · X − 1. What is E(Y )?

From the definition: Ωy = 2 · 1− 1, 2 · 0− 1 = 1,−1,each with probability 1/2, so

E(Y ) = −1 · 12

+ 1 · 12

= 0

Using the linearity of E:

E(Y ) = E(2X − 1) = 2E(X )− 1 = 2 · 12− 1 = 0

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Example

Let X be a Bernoulli(1/2) trial. We know E(X ) = p = 1/2. LetY = 2 · X − 1. What is E(Y )?

From the definition: Ωy = 2 · 1− 1, 2 · 0− 1 = 1,−1,each with probability 1/2, so

E(Y ) = −1 · 12

+ 1 · 12

= 0

Using the linearity of E:

E(Y ) = E(2X − 1) = 2E(X )− 1 = 2 · 12− 1 = 0

Albyn Jones Math 141

Example

Let X be a Bernoulli(1/2) trial. We know E(X ) = p = 1/2. LetY = 2 · X − 1. What is E(Y )?

From the definition: Ωy = 2 · 1− 1, 2 · 0− 1 = 1,−1,each with probability 1/2, so

E(Y ) = −1 · 12

+ 1 · 12

= 0

Using the linearity of E:

E(Y ) = E(2X − 1) = 2E(X )− 1 = 2 · 12− 1 = 0

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Expected Values scale like averages

The average height of an adult female in the US is about 64inches. What is the average height of an adult female in the USin centimeters?

The conversion factor is 2.54 cm per inch. Thus the averageheight of an adult female in the US is

2.54cmin× 64 in ≈ 162.5 cm

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More Properties of Expected ValueE is a linear operator

Let X and Y be RV’s, and a and b constants, then

E(a · X + b · Y ) = a · E(X ) + b · E(Y )

Thusthe expected value of a sum is the sum of the expectedvalues:

E(X + Y ) = E(X ) + E(Y )

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Expected Value of a Binomial

Let X be a Binomial(n,p) RV. From the definiton of expectation,

E(X ) =k=n∑k=0

k ·(

nk

)pkqn−k

which may be algebraically challenging for some. But we canthink of X as the sum of n (independent) Bernoulli(p) trials Yi ,each with E(Yi) = p, so

E(X ) = E(Y1 + . . .Yn) = E(Y1) + . . .E(Yn) = np

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Example: Coin Tossing

Toss a fair coin 100 times, and count the number of Heads.What is the expected number of Heads?

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The Poisson

Let X ∼ Poisson(µ). What is E(X )?

Hint: X is like a Binomial(n,p) with large n and small p, wherewe define µ = n · p.

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Summary

Expected value: a weighted average of possible outcomes.

E(X ) =∑

xipi

Linearity properties:

E(a + bX + cY ) = a + bE(X ) + cE(Y )

Expected value of X ∼ Binomial(n,p) is the sum ofexpected values of n Bernoulli(p) trials:

E(X ) = np

Albyn Jones Math 141