Transcript random variable

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 . It is a weighted average.
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
4
5
6
7
8
9
Mean of X (μx) =
V
1
2
3
Prob .301 .176 .125 .097 .079 .067 .058 .051 .046
Mean of V (μV) =
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 a rule for 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  .
You!
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 it would take
between 13 and 18 hours to
evacuate people living in a lowlying 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. DOES NOT WORK for standard deviations!
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.
Rule 1 for means: Find 3 + 2X and the new mean
Rule 2 for means: Find X + Y and mean of X + Y
Rule 1 for variances: Find 3 + 2X and the new variance
Example #1
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
Example #2
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.
Example #3
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 #4
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 #5
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  .