Math 141. Lecture 5: Expected Value. Albyn Jones 1. jones/courses/ Library 304. Albyn Jones Math 141

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1 Math 141 Lecture 5: Expected Value Albyn Jones 1 1 Library jones/courses/141

2 History The early history of probability theory is intimately related to questions arising in gambling. For example:

3 History The early history of probability theory is intimately related to questions arising in gambling. For example: Two gamblers playing a game that ends when one has won all of the other s money have to quit in the middle of the game. What is the fair division of the pot?

4 History The early history of probability theory is intimately related to questions arising in gambling. For example: Two gamblers playing a game that ends when one has won all of the other s money have to quit in the middle of the game. What is the fair division of the pot? What is the fair price of a lottery ticket?

5 Reading! READ CHAPTER 4!

6 Expected Value Definition: Let X be a random variable with sample space with corresponding probabilities Ω = {x 0, x 1, x 2,...} P(X = x 0 ) = p 0, P(X = x 1 ) = p 1, P(X = x 2 ) = p 2,... Then the expected value of X, denoted E(X) is E(X) = x 0 p 0 + x 1 p 1 + x 2 p 2... Expected value is just the weighted average of the set of possible outcomes, with weights equal to the probability each outcome occurs.

7 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

8 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

9 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 1 6 ) + (2 1 6 ) + (3 1 6 ) + (4 1 6 ) + (5 1 6 ) + (6 1 6 ) Adding it up: E(X) = 21 6 = 3.5

10 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k

11 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k Expectation by Definition: E(X) = (0 p q 0 ) + (1 p q 1 ) + (2 p q 2 ) +...

12 Example: X is Geometric(p) Probabilities: For k = 0, 1, 2,... P(X = k) = p q k Expectation by Definition: E(X) = (0 p q 0 ) + (1 p q 1 ) + (2 p q 2 ) +... A Math 112 exercise in summing an infinite series.

13 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six.

14 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads.

15 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads. 5 = 6 1 = 1 1/6 1 = 5/6 1/6 1 = 2 1 = 1 1/2 1 = 1/2 1/2

16 Expectation by Heuristic: Consider how long we wait for each six when rolling a fair die. In the long run, roughly 1 of every 6 rolls is a six. Thus the long run average must be 5 non-sixes before each six. Consider how long we wait for Heads when tossing a fair coin. In the long run, roughly 1 of every 2 tosses yields Heads. Thus the long run average is 1 Tail before each Heads. 5 = 6 1 = 1 1/6 1 = 5/6 1/6 1 = 2 1 = 1 1/2 1 = 1/2 1/2 1 p 1 = 1 p p = q p

17 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution.

18 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average...

19 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager.

20 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager. Average as typical value: The average family has 2.4 children.

21 Interpretation What does expected value mean? Physics analog: Center of mass of the probability distribution. In the long run: Sometimes more, sometimes less, but on the average... Gambling: The fair price of a wager. Average as typical value: The average family has 2.4 children. Silly question? How many families have 2.4 children?

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

23 Median: another measure of location Definition: Median Let X be a RV, then any number m satisfying and P(X m) 1 2 P(X m) 1 2 is a median. The median of a distribution may not be unique.

24 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10}

25 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3.

26 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3. Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 10}

27 Median: examples Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 5, 10} 3 satisfies the definition: at least half the population is less than or equal to 3, and at least half is greater than or equal to 3. Consider a population with values for some variable X What is the median? {1, 1, 2, 3, 4, 10} Any number in the interval [2, 3]!

28 Expectation as Typical Value For a symmetric, unimodal population, the mean and median agree, and both seem typical: Density Z

29 Expectation as Typical Value The mean is not so typical for a multimodal or skewed population: Density median mean X

30 Expectation as Typical Value: Income US Household Income: According to the US Census Bureau, in 2009 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 mean household income. The US has a very skewed income distribution. Depending on how you define income, The top 1% of the US population gets about 24% of all income, and the top 10% gets close to 50% of the total.

31 Properties of Expected Value Back 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 with respect to translation (shift) and rescaling.

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

33 Example Let X be a Bernoulli(1/2) trial. We know E(X) = p = 1/2. Let Y = 2 X 1. What is E(Y )? From the definition: Ω y = {2 1 1, 2 0 1} = {1, 1}, each with probability 1/2, so E(Y ) = = 0

34 Example Let X be a Bernoulli(1/2) trial. We know E(X) = p = 1/2. Let Y = 2 X 1. What is E(Y )? From the definition: Ω y = {2 1 1, 2 0 1} = {1, 1}, each with probability 1/2, so Using the linearity of E: E(Y ) = = 0 E(Y ) = E(2X 1) = 2E(X) 1 = = 0

35 Expected Values scale like averages The average height of an adult female in the US is about 64 inches. What is the average height of an adult female in the US in centimeters? The conversion factor is 2.54 cm per inch. Thus the average height of an adult female in the US is 2.54 cm in 64 in cm

36 More Properties of Expected Value E 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 ) Thus the expected value of a sum is the sum of the expected values: E(X + Y ) = E(X) + E(Y )

37 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 ( ) n p k q n k k which may be algebraically challenging for some. But we can think of X as the sum of n (independent) Bernoulli(p) trials Y i, each with E(Y i ) = p, so E(X) = E(Y Y n ) = E(Y 1 ) +... E(Y n ) = np

38 Example: Coin Tossing Toss a fair coin 100 times, and count the number of Heads. What is the expected number of Heads?

39 The Poisson Let X Poisson(µ). What is E(X)? Hint: X is like a Binomial(n,p) with large n and small p, where we define µ = n p.

40 Summary Expected value: a weighted average of possible outcomes. Linearity properties: E(X) = x i p i E(a + bx + cy ) = a + be(x) + ce(y ) Expected value of X Binomial(n, p) is the sum of expected values of n Bernoulli(p) trials: E(X) = np

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