Statistics – Wilsen
Unit 4: Probability
Chapter 16
Block 2
Discrete Random Variables: Center
Example 1
An insurance company offers a "death and disability" policy that pays $10,000
when you die, or $5000 if you are permanently disabled.
(a)
What is the random variable X here, and is it discrete or continuous?
(b)
Make a probability model for the payout. Actuarial tables tell you that 1 person
in 1000 dies each year, and 2 people in 1000 are disabled.
X = payout
(c)
x
P (X = x)
How much can the company expect to pay out ON AVERAGE, per policy?
Example 2
On Valentine’s Day, the Quiet Nook restaurant offers a Lucky Lovers Special that
could save couples money on their romantic dinners. When the waiter brings
the check, he’ll also bring the four aces from a deck of cards. He’ll shuffle
them and lay them out face down on the table. The couple will then get to turn
one card over. If it’s black, they’ll owe the full amount, but if it’s the ace of
hearts, the waiter will give them a $20 Lucky Lovers discount. If they first turn
over the ace of diamonds (hey – at least it’s red!), they’ll then get to turn over
one of the remaining cards, earning a $10 discount for finding the ace of hearts
this time. On average, how much can couples expect to save?
X = discount
x
P (X = x)
Example 3
One of the authors took his minivan in for repair recently because the air
conditioner was cutting out intermittently. The mechanic identified the
problem as dirt in a control unit. He said that in about 75% of such cases,
drawing down and then recharging the coolant a couple of times cleans up
the problem, and costs only $60. If that fails, then the control unit must be
replaced at an additional cost of $100 for parts and $40 for labor. What is the
expected value of the cost of this repair?
Discrete Random Variables: Spread
Example 4
The insurance company that offers a "death and disability" policy that pays
$10,000 when you die, or $5000 if you are permanently disabled. We already
found that the average cost to the insurance company is $20 per policy, though
of course, the company will never actually pay out $20 to anyone in particular.
What is the variance and standard deviation of all the possible payouts?
Example 5
In the Lucky Lovers restaurant, we found that couples can expect an average
savings of µ = $5.83 . Again, we need to realize that this will not be the savings
of any particular couple. What is the variance and standard deviation of the
savings?
Example 6… with your calculator
(a)
Create a probability model:
(b)
• type payouts into L1 and probabilities into L2
• go to 1-Var Stats, specifying both lists, and then look for µ and σ
Transforming and Combining Discrete Random Variables:
Expected Value
You get no points for rolling a 1, 2 or 3
You get 5 points for rolling a 4 or a 5
You get 50 points for rolling a 6
Find the expected value for each of the following:
(a)
points in this original game, E ( points )
(b)
now add 4 to all the original point values awarded, E ( points + 4 )
(c)
now double all the original point values awarded, E ( 2 • points )
(d)
you and your friend play the original game separately, calculating the sum of your
points, E ( points1 + points 2 )
(e)
you and your friend play the original game separately, calculating the difference of
your points, E ( points1 – points 2 )
Transforming and Combining Discrete Random Variables:
Variance
You get no points for rolling a 1, 2 or 3
You get 5 points for rolling a 4 or a 5
You get 50 points for rolling a 6
Find the variance for each of the following:
(a)
points in this original game, VAR ( points )
(b)
now add 4 to all the original point values awarded, VAR ( points + 4 )
(c)
now double all the original point values awarded, VAR ( 2 • points )
(d)
you and your friend play the original game separately, calculating the sum of your
points, VAR ( points1 + points 2 )
(e)
you and your friend play the original game separately, calculating the difference of
your points, VAR ( points1 – points 2 )
Example 7
Last year, each of the three AP stats classes happened to have a variance of
roughly 15 on the midterm exam.
(a) Ms. Wilsen decides to triple all of her students’ scores. What is the new
variance and standard deviation for her class?
(b) All three classes’ scores are combined. What is the variance and standard
deviation for all three classes?
When
X1
and
X2
are independent
VAR ( X1 + X2 ) = VAR ( X1 ) + VAR ( X2 )
Example 8
A delivery company’s trucks occasionally get parking tickets, and based on past
experience, the company plans that the trucks will average 1.3 tickets a month
each, with a standard deviation of 0.7 tickets. If they have 18 trucks, what are
the mean and standard deviation of the total number of parking tickets they’ll
have to pay this month?
Example 9
A farmer has 100 pounds of apples and 50 pounds of potatoes for sale. The
market price for apples (per pound) fluctuates each day with a mean of $0.50
and a standard deviation of $0.20. Similarly, for a pound of potatoes, the
mean price is $0.30 and the standard deviation is $0.10. It also costs the
farmer $2 to bring all the apples and potatoes to the market. The market is
busy with eager shoppers, so we can assume that he’ll be able to sell all of each
type of produce at that day’s price.
(a)
Define your variables, and use them to express the farmer’s net income.
(b)
Find the mean of the farmer’s net income. Clearly show your use of the
formulas.
(c)
Find the standard deviation of the net income. Clearly show your use
of the formulas.
Hours Asleep and Awake in the Last 24 Hours
Show that E(X + Y ) = E(X) + E(Y ) .
Show that Var(X + Y ) = Var(X) + Var(Y ) .
Continuous Random Variables: Center and Spread
Example 10
Consider a company that manufactures and ships small stereo systems. There
are two stages: packing the stereos and then boxing them. The time required
to pack the stereos can be described by a Normal model with a mean of 9
minutes and standard deviation of 1.5 minutes. Time for the boxing stage 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 the stereo systems takes longer to pack than to box?
AP problems