Today

1
Today

• Today Begin Chapter 3
• Reading
• Covered 2.1-2.5 from Chapter 2
• Please read Chapter 3
• Suggested Problems 3.2, 3.9, 3.12, 3.20, 3.23,
  3.24, 3R5, 3R9

2
Example

• Have 50 men, each 26 years old
• Have 50 women, each 28 years old
• What is the average age of the 100 people?

3
Example

• Have 500 men, each 26 years old
• Have 50 women, each 28 years old
• What is the average age of the 100 people?

4
Example

• Recall the game show, Lets Make a Deal
• A contestant had won 9,000 in prizes, and was
  offered to exchange the gifts for whatever lay
  behind one of three doors
• Behind one door was a 20,000 prize and behind
  the others were 5,000 and 2,000 prizes,
  respectively
• Should the contestant make the exchange?

5
Expectation (The Mean)

• If X is a discrete random variable with
  probability mass function f(x), the expected
  value (or mean value) is
• Provided the sum is absolutely convergent (if
  there are infinitely many values x1,x2, )
• Idea, the mean is the weighted average of the
  possible values of X

6
Example

• Recall the game show, Lets Make a Deal
• A contestant had won 9,000 in prizes, and was
  offered to exchange the gifts for whatever lay
  behind one of three doors
• Behind one door was a 20,000 prize and behind
  the others were 5,000 and 2,000 prizes,
  respectively
• Should the contestant make the exchange?

7
Example (True Story)

• When Derek was a graduate student in Vancouver,
  parking was 9.00/day
• If you parked illegally, the ticket was 10.00
• Derek discovered that he got a ticket about 50
  of the time
• Which is the better strategy
• Pay 9.00/day
• Park illegally

8
Properties of Expectation

• For random variables X and Y,
• E(c)c, where c is a constant
• E(cX)cE(X), where c is a constant
• E(XY) E(X) E(Y)
• E(aXbYc) aE(X) bE(Y)c, where a,b, and c
  are constants

9
Example

• Two dice are rolled one red and one green
• Let X be the outcome of the red die and Y be the
  outcome of the green die
• Find E(XY)

10
Conditional Mean

• Conditional distributions also have means
• The mean will be conditional on the value of
  another random variable
• The conditional mean of Y give Xx is

11
Example

• Three digits are picked at random, without
  replacement, from 1,2, , 8
• Let Y denote the largest digit and X denote the
  smallest
• Find the probability function for Y
• Find E(Y)
• Find probability function for Y given X3
• Find E(YX3)

12
Expected Value of a Function

• After observing a random variable, often
  interested in some function of the random
  variable
• The mean value of a function, g, of a random
  variable X is

13
Example (3.1)

• Consider a random variable X, with probability
  function f below
• Find E(X2)

x 0 1 2 3 4
f(x) .1 .3 .3 .1 .2
14
Example (3.10)

• Recall the game show, Lets Make a Deal
• A contestant had won 9,000 in prizes, and was
  offered to exchange the gifts for whatever lay
  behind one of three doors
• Behind one door was a 20,000 prize and behind
  the others were 5,000 and 2,000 prizes,
  respectively
• Suppose that the contestant owes a murderous loan
  shark 9000, due the next day
• Can use a utility function to help express
  thenotion of expectation in this situation

15
Expected Values and Joint Distributions

• Some useful relations
• How do we get these?

16
Expected Values and Joint Distributions

• When random variables X and Y are independent,
• How did we get this?