1. Chapter 6: Random Variables
Section 6.1
Discrete and Continuous Random Variables
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE

2. Chapter 6 Random Variables
Warm-up Page 355 #20 a) Make 2 histograms. One for the distribution of X and one for the distribution of Y. Draw them on your paper and describe any differences that you observe.
# of persons: 1 2 3 4 5 6 7
Household, px:
.25
.32
.17
.15
.07
.03
.01
# of persons: 1 2 3 4 5 6 7
Family, py:
.42
.23
.21
.09
.03
.02
Late flash cards & reading guides go on the cart in the “late tray”. Remember, any DIGITAL late work requires a full size note in the tray to let me know.
Today is the last day to turn in pennies.

3. Page 355 #20
X Y
# of persons: 1 2 3 4 5 6 7
Household, px:
.25
.32
.17
.15
.07
.03
.01
Family, py:
.42
.23
.21
.09
.02
a) Both distributions are skewed to the right. However P(X = 1) is much greater than P(Y = 1). This reflects the fact that a family must consist of 2 or more persons.

4. X Y
b) µX = 2.6 people and µY = 3.14 people. The family distribution has a slightly larger mean than the household distribution, probably because families cannot consist of just 1 person, whereas households can.
c) X = people and Y = people for a family. The standard deviation for households is slightly larger, mainly due to the fact there can be a 1 person household.

5. Do you have homework questions? R.E.D.
X Y
Page 355 #20
Do you have homework questions?
R.E.D.

6. Discrete and Continuous Random Variables
Discrete random variables commonly arise from situations that involve counting something. Situations that involve measuring something often result in a continuous random variable.
Discrete and Continuous Random Variables
Definition:
A continuous random variable X takes on all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.
The probability model of a discrete random variable X assigns a probability between 0 and 1 to each possible value of X.
A continuous random variable Y has infinitely many possible values. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability.

7. Example: Young Women’s Heights page 351
Read the example on page 351. Define Y as the height of a randomly chosen young woman. Y is a continuous random variable whose probability distribution is N(64, 2.7).
What is the probability that a randomly chosen young woman has height between 68 and 70 inches?
P(68 ≤ Y ≤ 70) = ???
P(1.48 ≤ Z ≤ 2.22) =
P(Z ≤ 2.22) – P(Z ≤ 1.48)
= – =
There is about a 5.6% chance that a randomly chosen young woman has a height between 68 and 70 inches.

8. Pg 356 #22 a) b) c)
P (Y < .4) = .4
P (Y < .4) = .4 .4

9. Pg 356 #22 a) P (Y < .4) = .4 b) P (Y < .4) = .4 c) .1 .15
P (.1 < Y < .15 or .77 < Y < .88)
= = .16

10. What is the probability that a randomly chosen student runs the mile in under 6 minutes?
The time Y of the randomly chosen student has the N(7.11, 0.74) distribution. We want to find P(Y<6). We’ll use Normalcdf to find the area under the Normal Curve.
Page 356 #24
State:
Plan:
Do:
Conclude:
6.0

11. P(y<6) = normcdf(-, 6, µ y = 7.11, y = .74) = .0668 State:
Page 356 #24
P(y<6) = normcdf(-, 6, µ y = 7.11, y = .74) = .0668
State:
Plan:
Do:
Conclude:
There is about a 7% chance that this student will run the mile in under 6 minutes.
6.0

12. Page 356 #26
N(9, .075) P (8.9 <  < 9.1) = normcdf(8.9, 9.1, µ x = 9, x = .075)
=.8176
Probability Rules

13. Section 6.2 Transforming and Combining Random Variables
Learning Objectives
After this section, you should be able to…
DESCRIBE the effect of performing a linear transformation on a random variable
COMBINE random variables and CALCULATE the resulting mean and standard deviation
CALCULATE and INTERPRET probabilities involving combinations of Normal random variables

15. Transforming and Combining Random Variables
Linear Transformations
In Section 6.1, we learned that the mean and standard deviation give us important information about a random variable. In this section, we’ll learn how the mean and standard deviation are affected by transformations on random variables.
Transforming and Combining Random Variables
In Chapter 2, we studied the effects of linear transformations on the shape, center, and spread of a distribution of data. Recall:
Adding (or subtracting) a constant, a, to each observation:
Adds a to measures of center and location.
Does not change the shape or measures of spread.
Multiplying (or dividing) each observation by a constant, b:
Multiplies (divides) measures of center and location by b.
Multiplies (divides) measures of spread by |b|.
Does not change the shape of the distribution.

16. Transforming and Combining Random Variables
Linear Transformations page 359
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 will hold up to 6 passengers. Define X as the number of passengers on a randomly selected day.
Transforming and Combining Random Variables
Passengers xi 2 3 4 5 6
Probability pi
0.15
0.25
0.35
0.20
0.05
The mean of X is 3.75 and the standard deviation is
Pete charges $150 per passenger. The random variable C describes the amount Pete collects on a randomly selected day.
Collected ci
300
450
600
750
900
Probability pi
0.15
0.25
0.35
0.20
0.05
The mean of C is $ and the standard deviation is $
Compare the shape, center, and spread of the two probability distributions.

17. Transforming and Combining Random Variables
Linear Transformations
How does multiplying or dividing by a constant affect a random variable?
Transforming and Combining Random Variables
Effect on a Random Variable of Multiplying (Dividing) by a Constant
Multiplying (or dividing) each value of a random variable by a number b:
Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b.
Multiplies (divides) measures of spread (range, IQR, standard deviation) by |b|.
Does not change the shape of the distribution.
Note: Multiplying a random variable by a constant b multiplies the variance by b2.

18. µH = 5.8 feet; H = .24 feet
µJ = (5.8 ft)(12 in./1 ft) = 69.6 in.
J = (.24 ft)(12 in./1ft) = 2.88 in.
Page 378 #36

19. Transforming and Combining Random Variables
Linear Transformations
Consider Pete’s Jeep Tours again. We defined C as the amount of money Pete collects on a randomly selected day.
Transforming and Combining Random Variables
Collected ci
300
450
600
750
900
Probability pi
0.15
0.25
0.35
0.20
0.05
The mean of C is $ and the standard deviation is $
It costs Pete $100 per trip to buy permits, gas, and a ferry pass. The random variable V describes the profit Pete makes on a randomly selected day.
Profit vi
200
350
500
650
800
Probability pi
0.15
0.25
0.35
0.20
0.05
The mean of V is $ and the standard deviation is $
Compare the shape, center, and spread of the two probability distributions.

20. Transforming and Combining Random Variables
Linear Transformations
How does adding or subtracting a constant affect a random variable?
Transforming and Combining Random Variables
Effect on a Random Variable of Adding (or Subtracting) a Constant
Adding the same number a (which could be negative) to each value of a random variable:
Adds a to measures of center and location (mean, median, quartiles, percentiles).
Does not change measures of spread (range, IQR, standard deviation).
Does not change the shape of the distribution.

21. Transforming and Combining Random Variables
Linear Transformations
Whether we are dealing with data or random variables, the effects of a linear transformation are the same.
Transforming and Combining Random Variables
Effect on a Linear Transformation on the Mean and Standard Deviation
If Y = a + bX is a linear transformation of the random variable X, then
The probability distribution of Y has the same shape as the probability distribution of X.
µY = a + bµX.
σY = |b|σX (since b could be a negative number).

22. Page 378 #38
This distribution is skewed to the right. Ana is more likely to get 10 or 20 points than 40 or 50 points.
µS = 23.8; S = 12.63
µT = 2.38; T = 1.263
Score 10 20 30 40 50
Probability
.32
.27
.19
.15
.07
Tickets 1 2 3 4 5
Probability
.32
.27
.19
.15
.07

23. b) On average, Ana will receive 2.38 tickets on a single roll.
µS = 23.8; S = 12.63
µT = 2.38; T = 1.263
#38)
Score 10 20 30 40 50
Percentile
.32
.27
.19
.15
.07
This distribution is skewed to the right. Ana is more likely to get 1 or 2 tickets than 4 or 5.
c) On individual rolls, the number of tickets that Ana wins will vary by from the mean (2.38) on average.
Tickets 1 2 3 4 5
Percentile
.32
.27
.19
.15
.07
b) On average, Ana will receive tickets on a single roll.

24. N Mean Median St.Dev Min Max Q1 Q3
Distribution of number of correct answers
Page 378 #40
b) IQR = 10(9) – 10(8) = 10
a) Median = 10(8.5) = 85
Since the distance between the median and the minimum is much larger than the distance between the median and the maximum, this distribution is skewed to the left. The shape of G’s distribution will be the same as the shape of X’s distribution.

25. µX = 0(.999) + 500(.001) = $.50
(X)2 = (0-.5) 2(.999) + (500-.5) 2(.001) =
Page 378 #42
b) µW = µX – 1 = .5 – 1 = -$ W = X = $15.80
() = $15.80
Y = .9X - .2
44. µX = 3; X = 2.52
µY = .9 µX - .2
Y = .9 X
= $2.268 million
= = $2.5 million

26. X ~ N(9.7, .03) ounces µY = 9.7 – 9.63 = .07 ounces Y = .03 ounces
Transform to grams by multiplying by µY = grams Y = grams
Page 379 #46
Y ~ N(1.985, .8505) g
P(y>3) = normcdf(3, , µ Y = 1.985, Y = .8505)
= .1170

27. Read pg
Do pg 355 #15-25 odd
Do pg 378 #35-45 odd
Video 6.2 (9 min)
Reading Guide 6.2 #1-5
Chapter 6 Vocabulary Flash Cards
Estimated Chapter 6 Test Date is Friday March 4.

28. Section 6.1 Discrete and Continuous Random Variables
Summary
In this section, we learned that…
The mean of a random variable is the long-run average value of the variable after many repetitions of the chance process. It is also known as the expected value of the random variable.
The expected value of a discrete random variable X is
The variance of a random variable is the average squared deviation of the values of the variable from their mean. The standard deviation is the square root of the variance. For a discrete random variable X,

29. Looking Ahead… In the next Section…
We’ll learn how to determine the mean and standard deviation when we transform or combine random variables.
We’ll learn about
Linear Transformations
Combining Random Variables
Combining Normal Random Variables
In the next Section…