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POD #50
12/2/2011
2007B #6b
Is this model
appropriate?
Write the
equation of
the LSRL
Stats: Modeling the World
Chapter 16
Random Variables
Greedy Pig game
What is a Random Variable?
A random variable assumes a value based on
the outcome of a random event.
•We use a capital letter, like X, to denote a
random variable.
•A particular value of a random variable will
be denoted with the corresponding lower
case letter, in this case x.
Two Types of Random Variables:
Discrete random variables can take one of a
countable number of distinct outcomes.
Example: Number of credit hours
Continuous random variables can take any
numeric value within a range of values.
Example: Cost of books this term
Measure of Center: Expected Value
  E ( X )   xP( x)
Measure of Center: Expected Value
Policyholder
Outcome
Death
Disability
Neither
Payout
x
10,000
5,000
0
Probability
P(X=x)
1/1000
2/1000
997/1000
Find the expected value of the insurance company payout E(X):
Measure of Spread: Standard Deviation
  Var ( X )   ( x   ) P( x)
2
  SD( X )  Var ( X )
2
Measure of Spread: Standard Deviation
Policyholder
Outcome
Death
Disability
Neither
Payout
x
10,000
5,000
0
Probability
P(X=x)
1/1000
2/1000
997/1000
Deviation
(X-µ)
(10,000-20)=9980
(5000-20)=4980
(0-20)=-20
Find the standard deviation of the insurance company payout SD(X) or σ:
Example:
A carnival game offers a $100 cash prize for anyone who can break a
balloon by throwing a dart at it. It costs $5 to play, and you’re willing to
spend up to $20 trying to win. You estimate that you have about a 10%
chance of hitting the balloon on any throw.
a) Create a probability model for this carnival game.
b) Find the expected number of darts you will throw.
c) Find your expected winnings.
d) Find the standard deviation of your winnings.
.
Example:
.
A carnival game offers a $100 cash prize for anyone who can break a
balloon by throwing a dart at it. It costs $5 to play, and you’re willing to
spend up to $20 trying to win. You estimate that you have about a 10%
chance of hitting the balloon on any throw.
a) Create a probability model for this carnival game.
Net Winnings
$95
$90
Number of Darts 1 dart 2 darts
P(Amount won) =0.1 (0.9)(0.1)
=0.09
$85
3 darts
(0.9)(0.9)(0.1)
=0.081
$80
4 darts (win)
(0.9)3(0.1)
=0.0729
-$20
4 darts (lose)
(0.9)4
=0.6561
Net Winnings
$95
$90
Number of Darts 1 dart 2 darts
P(Amount won) =0.1 (0.9)(0.1)
=0.09
$85
3 darts
(0.9)(0.9)(0.1)
=0.081
$80
4 darts (win)
(0.9)3(0.1)
=0.0729
-$20
4 darts (lose)
(0.9)4
=0.6561
A carnival game offers a $100 cash prize for anyone who can break a
balloon by throwing a dart at it. It costs $5 to play, and you’re willing to
spend up to $20 trying to win. You estimate that you have about a 10%
chance of hitting the balloon on any throw.
b) Find the expected number of darts you will throw.
Net Winnings
$95
$90
Number of Darts 1 dart 2 darts
P(Amount won) =0.1 (0.9)(0.1)
=0.09
$85
3 darts
(0.9)(0.9)(0.1)
=0.081
$80
4 darts (win)
(0.9)3(0.1)
=0.0729
-$20
4 darts (lose)
(0.9)4
=0.6561
A carnival game offers a $100 cash prize for anyone who can break a
balloon by throwing a dart at it. It costs $5 to play, and you’re willing to
spend up to $20 trying to win. You estimate that you have about a 10%
chance of hitting the balloon on any throw.
c) Find your expected winnings.
Net Winnings
$95
$90
Number of Darts 1 dart 2 darts
P(Amount won) =0.1 (0.9)(0.1)
=0.09
$85
3 darts
(0.9)(0.9)(0.1)
=0.081
$80
4 darts (win)
(0.9)3(0.1)
=0.0729
-$20
4 darts (lose)
(0.9)4
=0.6561
A carnival game offers a $100 cash prize for anyone who can break a
balloon by throwing a dart at it. It costs $5 to play, and you’re willing to
spend up to $20 trying to win. You estimate that you have about a 10%
chance of hitting the balloon on any throw.
d) Find the standard deviation of your winnings.
 2  Var (Winnings)
 (95  17.20)2 (0.1)  (90  17.20)2 (0.09)  (85  17.20)2 (0.081)  (80  17.20)2 (0.0729)  (20  17.20)2 (0.6561)
 2650.057
  SD(Winnings)  Var (Winnings)  2650.057  $51.48
Try This:
.
A small software company bids an two contracts. It anticipates a profit of
$50,000 if it gets the larger contract and a profit of $20,000 on the smaller
contract. The company estimates there’s a 30% chance it will get the
larger contract and a 60% chance it will get the smaller contract.
Assuming the contracts will be awarded independently.
a) Organize the events in a Venn diagram.
b) Create a probability model for the company.
c) What is the expected profit?
d) What is the standard deviation of the profit?
Try This:
.
A small software company bids an two contracts. It anticipates a profit of
$50,000 if it gets the larger contract and a profit of $20,000 on the smaller
contract. The company estimates there’s a 30% chance it will get the
larger contract and a 60% chance it will get the smaller contract.
Assuming the contracts will be awarded independently.
a) Organize the events in a Venn diagram.
Larger
0.12 0.18
Smaller
0.42
0.28
A small software company bids an two contracts. It anticipates a profit of
$50,000 if it gets the larger contract and a profit of $20,000 on the smaller
contract. The company estimates there’s a 30% chance it will get the
larger contract and a 60% chance it will get the smaller contract.
Assuming the contracts will be awarded independently.
b) Create a probability model for the company.
Profit
P(Profit)
Larger Only Smaller Only Both
$50,000
$20,000
$70,000
0.12
0.42
0.18
Neither
$0
0.28
Profit
P(Profit)
Larger Only Smaller Only Both
$50,000
$20,000
$70,000
0.12
0.42
0.18
Neither
$0
0.28
A small software company bids an two contracts. It anticipates a profit of
$50,000 if it gets the larger contract and a profit of $20,000 on the smaller
contract. The company estimates there’s a 30% chance it will get the
larger contract and a 60% chance it will get the smaller contract.
Assuming the contracts will be awarded independently.
c) What is the expected profit?
  E (Profit)=$50,000(0.12)+$20,000(0.42)+$70,000(0.18)+$0(0.28)
=$27,000
d) What is the standard deviation of the profit?
 2  Var (Profit)=($50,000-$27,000) 2 (0.12)  ($20,000-$27,000)2 (0.42)  ($70,000-$27,000) 2 (0.18)  ($0-$27,000)2 (0.28)
 621,000,000
  Var (Profit)  621, 000, 000  $24,919.87
More about Means and Variances
Adding or subtracting a constant from data shifts
the mean but doesn’t change the variance or
standard deviation:
E(X ± c) = E(X) ± c
Var(X ± c) = Var(X)
Example: Consider everyone in a company
receiving a $5000 increase in salary.
More about Means and Variances
In general, multiplying each value of a random
variable by a constant multiplies the mean by that
constant and the variance by the square of the
constant:
E(aX) = aE(X)
Var(aX) = a2Var(X)
Example: Consider everyone in a company
receiving a 10% increase in salary.
More about Means and Variances
In general,
The mean of the sum of two random variables is the
sum of the means.
The mean of the difference of two random variables is
the difference of the means.
E(X ± Y) = E(X) ± E(Y)
If the random variables are independent, the variance
of their sum or difference is always the sum of the
variances.
Var(X ± Y) = Var(X) + Var(Y)
*Standard Deviations DO NOT add!!!
Example:
.
Couples dining at the Quiet Nook can expect Lucky Lovers discounts
averaging $5.83 with a standard deviation of $8.62.
a) Suppose that for several weeks the restaurant has also been
distributing coupons worth $5 off any one meal (one discount per
table). If every couple dining there on Valentine’s Day bring a coupon,
what will the mean and standard deviation of the total discounts they’ll
receive?
Example:
.
Couples dining at the Quiet Nook can expect Lucky Lovers discounts
averaging $5.83 with a standard deviation of $8.62.
b) When two couples dine together on a single check the restaurant
doubles the discount offer. What are the mean and standard deviation
of discounts for the foursome?
Example:
.
Couples dining at the Quiet Nook can expect Lucky Lovers discounts
averaging $5.83 with a standard deviation of $8.62.
c. Now two couple are dining together but decide to get separate checks.
What is the mean and standard deviation for this situation?
Example:
.
Couples dining at the Quiet Nook can expect Lucky Lovers discounts
averaging $5.83 with a standard deviation of $8.62.
d. Now there is another restaurant called the Wise Fool that has a
competing discount. The discount at the wise fool has an average of
$10 and a standard deviation of $15. How much more can you expect
to save at the Wise Fool and with what standard deviation?
Try This:
.
A casino knows that people play the slot machines in hopes of hitting the
jackpot but that most of them lose their dollar. Suppose a certain machine
pays out an average of $0.92, with a standard deviation of $120.
a) Why is the standard deviation so large?
b) If you play 5 times, what are the mean and standard deviation of the
casino’s payout?
c) If gamblers play this machine 1000 times in a day, what are the mean
and standard deviation of the casino’s payout?
Try This:
.
The American Veterinarian Association claims that the annual cost of
medical care for dogs averages $100, with a standard deviation of $30,
and for cats averages $120, with a standard deviation of $35.
a) What’s the expected difference in the cost of medical care for dogs and
cats?
b) What’s the standard deviation of the difference?
Example:
.
A company manufacturer small stereo systems. At the end of the
production line, the stereos are packaged and prepared for shipping. Stage
1 is called “packing” can be described by the Normal model with a mean of
9 minutes and a standard deviation of 1.5 minutes. The second stage is
called “boxing” which can also be modeled as Normal with a mean of 6
minutes and standard deviation of 1 minute.
a) What is the probability that packing two consecutive systems takes over
20 minutes?
b) What percentage of stereo systems take longer to pack than to box?
Example:
.
A company manufacturer small stereo systems. At the end of the
production line, the stereos are packaged and prepared for shipping. Stage
1 is called “packing” can be described by the Normal model with a mean of
9 minutes and a standard deviation of 1.5 minutes. The second stage is
called “boxing” which can also be modeled as Normal with a mean of 6
minutes and standard deviation of 1 minute.
a) What is the probability that packing two consecutive systems takes over
20 minutes?
b) What percentage of stereo systems take longer to pack than to box?