Download Linear Combinations Day 1

Combinations Day 1
Name: ______________________________
Let’s investigate a dice problem. First, find (or recall) the expected value and standard deviation for the roll of one die.
E(X) = ______
X
= ____________
Now…Option 1 is to roll one die and double its value, obtaining 2, 4, 6, 8,10, or 12. Option 2 is to roll two dice and find their sum,
obtaining 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. Option 3 is to roll two dice and find their difference obtaining -5, -4, -3, -2, -1, 0, 1, 2, 3,
4, or 5.
a. Write the probability distribution of option 1. Find the mean and standard deviation of the distribution.
E(2X) = ______
 2 X = ____________
b. Write the probability distribution of option 2. Find the mean and standard deviation of the distribution.
E(X+X) = ______
 X  X = ____________
c. Write the probability distribution of option 3. Find the mean and standard deviation of the distribution.
E(X - X) = ______
 X  X = ____________
d. Compare the means and standard deviation for all three options.
e. Come up with a formula for the mean and standard deviation for Option 1, Option 2, and Option 3 in terms of the mean and
standard deviation for the roll of one die.
E(2X) = _________________
 2 X = _________________
E(X - X) = _________________
E(X+X) = _________________
 X  X = _________________
 X  X = _________________
If you have just one random variable and you add/subtract a number to the list of data or multiply the entire list by a constant, then you
have performed a linear transformation.
If you are combining more than one random variable then you are doing a combination, and you should use the appropriate formulas.
Ex 2: Suppose that E(X) = 2.5, VAR(X) = .16, E(Y) = 1.2, VAR(Y) = .36, and X and Y are independent.
a) What is E(X+Y),  ( X  Y ) , and VAR(X+Y)?
b) What is E(X-Y),
 ( X  Y ) and VAR(X-Y)?
c) What is E(X+X) and  ( X  X ) ?
d) What is E(2X-4Y),
 (2 X  4Y ) ?
Rules for Combinations
1. E(X+Y) =
E(X-Y) =
2.
 2(X Y) =
 2(X Y) =
3.
 (X Y) =
 (X Y)
Assignment
1. Insurance companies use statistics to analyze how much money policy holders should pay in order to payout on claims made while
maintaining a profit for the owners. For simplicity:
Insurance Company StayAlive insures 10,000 women (all approximately the same age, good health, etc) for a policy worth a payout of
$100,000 if policyholder dies or $50,000 if permanently disabled. Based on the insurance company’s analysis, they expect only one
woman to pass away and two to become permanently disabled out of the 10,000 policy holders.
a) What is the insurance company’s expected payout?
b) What is the variance of payout in this situation?
Policyholder
Payout, Y
Probability
c) What is the standard deviation of payout?
outcome
P(Y = y)
d) What is the expected payout and standard deviation if the company doubles
Death
$100,000
1
the payout for anyone of the 10,000 women policyholders?
10000
e) Assuming each policy is independent, what would be the total expected
Disabled
$50,000
payout and standard deviation for any two of the women policyholders?
2
Neither
10000
9997
$0
10000
2. On your drive (or ride) to school you have to pass through a total
R=# of red lights
0
of 3 traffic lights. Assuming you stop when the light is red, the
P(R = r)
.05
probability model of the number of stops needed and likelihood of
number of red lights is shown in the table.
a) How many red lights should you expect to “hit” each day?
b) What’s the standard deviation?
Assuming each day is independent:
c) What is the expected number of red lights to hit on your way to school during a 5-day school week?
d) What’s the standard deviation for this five day time span?
1
.20
2
.30
3
.45
3. Theif! There are 12 batteries on Mrs. Gann’s desk, unknowing to you 4 of the batteries are dead. You steal a random sample of 2
batteries for your Silver calculator.
a) Create a tree diagram representing the situation described.
b) Create a probability model for the number of good batteries you get.
c) What’s the expected number of good ones you have stolen?
d) What’s the standard deviation?
4. Fire! An insurance company estimates that it should make an annual profit of $150 on each homeowner’s policy written with a
standard deviation of $6000. Assume each policy is independent of each other.
a) Why is the standard deviation so large?
b) If the company writes only two of these policies, what are the mean and standard deviation of the annual profit?
c) If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit?
d) Do you think the company is likely to be profitable? Explain. Hint: draw and label a normal curve with the information calculated
in part (c).
e) What assumptions underlie your analysis?
Can you think of circumstances under which the assumption might be violated? Explain.
5. A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples (per pound) each day is a random variable
with a mean of $0.50 and a SD of $0.20. Similarly, for a pound of potatoes, the mean price is $0.30 and the SD is $0.10. It also costs
him $2.00 to bring all the produce to the market. Assume that he’ll be able to sell all of each type of produce at the day’s price.
a) Define your random variables.
b) Use your defined random variables in part (a) to express the farmer’s net income.
c) Find the mean of the farmer’s net income.
d) Find the standard deviation of the farmer’s net income.
e) Do you need to make any assumptions in calculating the mean?
f) Do you need to make any assumptions in calculating the standard deviation?