Download 7.2

X
The mean of a set of observations is their ordinary
average, whereas the mean of a random variable X is
an average of the possible values of X
The mean of a random variable X is often called the
expected value of X, and describes the long-run
average outcome.
Example 7.6, p. 483
X
1
2
3
4
5
6
7
8
9
Prob
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
V
1
2
3
4
5
6
7
8
9
Prob .301 .176 .125 .097 .079 .067 .058 .051 .046
Locating the Mean in a discrete
distribution = the center of the
distribution
Example (Mean of a prob.
distribution = Expected Value)
Two dice are rolled simultaneously. If
both show a 6, then the player wins
$20, otherwise the player loses the
game. It costs $2.00 to play the game.
What is the expected gain or loss of the
game?
Ex. 7.7
Linda sells cars and motivates herself
by using probability estimates of her
sales. She estimates her car sales as
follows:
Cars sold
0
1
2
3
Prob.
.3
.4
.2
.1
Find the mean and variance of X.
Ex. 7.8: The distribution
of the heights of
women is close to
normal with N(64.5,
2.5). The graph plots
the values as we add
women to our sample.
Eventually the mean of
the observations gets
close to the population
mean and settles down
at that value.
More on Law of Large #’s
 The law says that the average results of many
independent observations/decisions are stable and
predictable. Insurance companies, grocery stores, and
other industries can predict demand even though their
many customers make independent decisions.
A grocery store deciding on how many gallons of milk to
stock
A fast food restaurant deciding how many beef patties to
prepare
Gambling casinos (games of chance must be variable in
order to hold the interest of gamblers). Even a long
evening in a casino is unpredictable; the house plays often
enough to rely on the law of large numbers, but you
don’t. The average winnings of the house on tens of
thousands of bets will be very close to the mean of the
distribution of winnings (guaranteeing a profit).
How large is a large number?
Can’t write on a rule on how many trials
are needed to guarantee a mean
outcome close to  ; this depends on
the variability of the random outcomes.
The more variable the outcomes, the
more trials are needed to ensure that
the mean outcome X is close to the
distribution mean  .
Emergency Evacuation
A panel of meteorological and
civil engineers studying
emergency evacuation plans
for Florida’s Gulf Coast in the
event of a hurricane has
estimated that is would take
between 13 and 18 hours to
evacuate people living in a
low-lying land, with the
probabilities shown here.
Find the mean, variance, and
standard deviation of the
distribution.
Time to
Probability
Evacuate
(nearest hr)
13
0.04
14
0.25
15
0.40
16
0.18
17
0.10
18
0.03
To find the sum (or difference) of the means of 2 random
variables, add the individual means of the random variables
X and Y together.
If 2 random variables are independent, the variance of the
sum (or difference) of the 2 random variables is equal to the
sum of the 2 individual variances.
Example
The following data comes from a normally
distributed population. Given that
X={2, 9, 11, 22} and Y={5, 7, 15,
21}, illustrate the rules for means and
variances.
Example
Gain Communication sells units to both the military
and civilian markets. Next year’s sales depend on
market conditions that are unpredictable. Given the
military and civilian division estimates and the fact
that Gain makes a profit of $2000 on each military
unit sold and $3500 on each civilian unit sold, find:
a) The mean and the variance of the number X of
communication units.
b) The best estimate of next year’s profit.
Military
Units sold: 1000 3000 5000 10,000
Probability: .1 .3
.4
.2
Civilian
Units sold: 300 500 750
Probability: .4
.5
.1
The probabilities that a randomly selected customer
purchases 1, 2, 3, 4, or 5 items at a convenience store
are .32, .12, .23, .18, and .15, respectively.
a) Construct a probability distribution (table) for the data
and verify that this is a legitimate probability distribution.
b) Calculate the mean of the random variable. Interpret
this value in the context of this problem.
c) Find the standard deviation of X.
d) Suppose 2 customers (A and B) are selected at
random. Find the mean and the standard deviation of
the difference in the number of items purchased by A
and by B. Show your work.
Any linear combination of independent Normal random
variables is also Normally Distributed.
Suppose that the mean height of
policemen is 70 inches w/a standard
deviation of 3 inches. And suppose that
the mean height for policewomen is 65
inches with a standard deviation of 2.5
inches. If heights of policemen and
policewomen are Normally distributed,
find the probability that a randomly
selected policewoman is taller than a
randomly selected policeman.
Example
Here’s a game: If a player rolls two dice and
gets a sum of 2 or 12, he wins $20. If the
person gets a 7, he wins $5. The cost to
play the game is $3. Find the expected
payout for the game.
Example
The random variable X takes the two values   
and    , each with probability 0.5. Use the
definition of mean and variance for discrete
random variables to show that X has mean  and
standard deviation  .
Airlines routinely overbook flights because they expect a certain
number of no-shows. An airline runs a 5 P.M. commuter flight from
Washington, D.C. to NYC on a plane that holds 38 passengers. Past
experience has shown that if 41 tickets are sold for the flight, then
the probability distribution for the number who actually show up
for the flight is shown in the table below:
# Who
actually
show
up
36
Prob. 0.46
37
38
39
40
41
0.30
0.16
0.05
0.02
0.01
Assume that 41 tickets are sold for each flight.
(a) There are 38 passenger seats on the flight. What is the
probability that all passengers who show up for this flight will
get a seat?
(b) What is the expected number of no-shows for this flight?
(c) Given that not all passenger seats are filled on a flight, what is
the probability that only 36 passengers showed up for the
flight?