Probability Distributions and Expected Value

1
Probability Distributions and Expected Value
2

• In previous chapters, our emphasis was on the
  probability of individual outcomes.
• This chapter develops models for distributions
  that show the probabilities for all possible
  outcomes of an experiment.

3
Random Variable (X)

• Has a single value, x, for each outcome of an
  experiment.
• To show all the possible outcomes, a chart is
  normally used.

4
Discrete variables take values that are separate
(or that can be counted)Continuous variables
have an infinite number of possible outcomes
(usually measurements that can have an unlimited
decimal place)
5
For example

• The number of phone calls made by a salesperson
• Discrete (1,2,3,4,5..)

6
For example

• The length of time a person spends on the phone
• Continuous (1 min, 1.23min ..)

7
Let a DRV (X) be the possible outcomes when
rolling a die

• The probability distribution could be written in
  a table

This is a uniform distribution, because all the
probabilities are the same.
8
A graph would look like this
1/6

• 1

2
3
4
5
6
9
Remember the PD for the sums generated by rolling
2 dice?

• 7

10
Expected Value Informal

• When rolling 2 dice, the sum that is generated
  most frequently is called the expected value. (7)
• This can also be calculated mathematically.
• Multiply each roll by its probability of
  occuring

11

• E(sum) 2P(sum 2) 3P(sum 3)
• 12P(sum 12)
• 2 X 1 / 36 3 X 2 / 36
• 252 / 36
• 7

12
The expected value, E(X), is the predicted
average of all possible outcomes.It is equal to
the sum of the products of each outcome with its
probability
13
Expected Value of a Discrete Random Variable

• The sum of the terms of the form (X)(PX)

E(X) x1P(X x1) x2P(X x2) xnP(X
xn)
14
Ex 1 Suppose you toss 3 coins.

• What is the likelihood that you would observe at
  least two heads?
• What is the expected number of heads?

15
Represent the theoretical probability
distribution as a table. The DRV, X, represents
the number of heads observed.

• X 0 hs 1 hs 2 hs 3 hs

P(X) x
16

• a) P(X 2) P(X 3) 3 / 8 1 / 8
• 1 / 2
• b) The expected number of heads
• 0(1 / 8) 1(3 / 8) 2(3 / 8) 3(1 / 8)
• 3 / 2

17
For a game to be fair, E(X) must be zero Consider
a dice game

• If you roll a 1 2 3 you win 1.00
• If you roll a 4 5 6 you pay 1.00
• Is this game fair?
• E(X) (1)(1/6) (1)(1/6) (1)(1/6) (-1)(1/6)
  (-1)(1/6) (-1)(1/6)

18

• Page 374
• 1, 2(ex 2), 3a,c, 4, 9, 11,12, 19