Download Transforming Random Variables Examples

Section 6.2 Part #1: Transforming Random Variables
AP Statistics
Name: ___________________
Date: ___________ Period: __
Scenario: Pete’s Jeep Tours offers a popular half-day trip in a tourist area. There must be at least 2 passengers
for the trip to run, and the vehicle can hold up to 6 passengers. The number of passengers X on a randomly
selected day has the following probability distribution.
No. of passengers xi
Probability pi
2
3
4
5
6
0.15
0.25
0.35
0.20
0.05
a) Draw a histogram of the probability distribution of X. Describe the shape of the histogram.
b) Find the mean of X and interpret it. Then find the standard deviation of X and interpret it.
𝜇𝑥 = 𝐸(𝑋) = __________
𝜎𝑥2 = __________
𝜎𝑥 = __________
Scenario #2: Pete charges $150 per passenger. Let C = the total amount of money that Pete collects on a
randomly selected trip. Because the amount of money Pete collects from the trip is just $150 times the number
of passengers, we can write the equation C = 150X. From the probability distribution of X, we can see that the
chance of having two people (X = 2) on the trip is 0.15. In that case, C = 150(2) = 300. So one possible value of C
is $300, and its corresponding probability is 0.15. If X = 3, then C = 150(3) = 450, and the corresponding
probability is 0.25. Fill in the remaining boxes of probability distribution of C.
Total Collected ci
300
450
Probability pi
0.15
0.25
0.35
0.20
0.05
a) Draw a histogram of the probability distribution of C. Describe the shape.
b) Find the mean of C and the standard deviation of C.
𝜇𝑐 = 𝐸(𝐶) = __________
𝜎𝑐2 = _________
𝜎𝑐 = __________
Summarize: Multiplying/Dividing a Random Variable by b
 It will multiply/divide the mean by ________
 It will multiply/divide the standard deviation by ________
 It will multiply/divide the variance by __________
 The shape of the distribution will _______________________________.
 Can you recall what it will do to the median and quartiles? Can you recall what it will do to the
range? (Think back to Chapter 2)
Scenario #3: It costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit
V that Pete makes from the trip is the total amount of money C that he collects from the passengers minus
$100. That is, V = C – 100. If Pete has only two passengers on the trip (X = 2), then C = 300 and V = $200. From
the probability distribution of C, the chance that this happens is 0.15. So the smallest possible value of V is
$200; its corresponding probability is 0.15. If X = 3, then C = 450 and V = 350, and the corresponding
probability is 0.25. Fill in the remaining boxes of probability distribution of V.
Total Collected vi
200
350
Probability pi
0.15
0.25
0.35
0.20
0.05
a) Draw a histogram of the probability distribution of V. Describe the shape.
b) Find the mean of V and the standard deviation of V.
𝜇𝑣 = 𝐸(𝑉) = __________
𝜎𝑣2 = _________
𝜎𝑣 = __________
Summarize: Adding/Subtracting a Random Variable by a
 It will add/subtract ______ to the mean.
 How will it affect the variance and standard deviation?
 The shape of the distribution will _______________________________.
 Can you recall what it will do to the median and quartiles? Can you recall what it will do to the
range? (Think back to Chapter 2)