AP Statistics
Chapter 6 Outline
Name:_____________________________
Calendar:
Mon 11/2: 6.1 Discrete and Continuous Random Variables
HW: 6.1 #1,5,7,9,13,14,18,19,23,25
Weds 11/4: 6.2 Transforming and Combining Random Variables
HW: 6.2 #37, 39-41, 43, 45
Mon 11/9: 6.2 Transforming and Combining Random Variables
HW: 49, 51, 57-79, 61, 63
Fri 11/13: 6.3 Binomial and Geometric Random Variables
HW: 71, 73, 75, 77, 93, 95, 97, 99, 101-103
Mon 11/16: Review Chapter 6
Weds 11/18: Chapter 6 Test
6.1 Discrete and Continuous Random Variables
Probability model
Random variable
Discrete Random Variables:
To find the mean (expected value)…
To find the standard deviation…
Probability distribution
Continuous Random Variables:
Example 2 of Discrete Random Variable:
Johnny is running the balloon darts game at the school fair. He has blown up hundreds of balloons with
notes about prize tickets inside them. Twelve percent of the notes say “You win 5 tickets,” twenty percent
say “You win 3 tickets,” and the rest say “Sorry, try again!” After each play, he replaces the popped
balloon with another one bearing the same note. Let T = the number of tickets won by a randomly selected
player of this game.
a. Give a probability distribution for this situation
b. Find and interpret the expected value
c. Find and interpret the standard deviation
Example of Continuous Random Variable:
The heights of young women closely follow the Normal distribution with 𝑁[64, 2.7]. Choose one young
women at random. Call her height Y. If we repeat the random choice many times, the distribution of
values of Y is the same Normal distribution that describes the heights of all young women.
a. Find the probability that the chosen women is between 68 and 70 inches tall.
6.2 Transforming and Combining Random Variables
Effect on a Random Variable of Multiplying or Dividing by a Constant
•
•
•
____________ measures of center and location (mean, median, quartiles, percentiles) by b.
____________ measures of spread (range, IQR, standard deviation) by |b|.
____________the shape of the distribution
Effect on a Random Variable of Adding or Subtracting by a Constant
•
•
•
____________measures of center and location (mean, median, quartiles, percentiles).
____________measures of spread (range, IQR, standard deviation).
____________the shape of the distribution.
Example:
In a large introductory statistics class, the distribution of X = raw scores on a test was approximately
normally distributed with a mean of 17.2 and a standard deviation of 3.8. The professor decides to scale
the scores by multiplying the raw scores by 4 and adding 10.
a. Define the variable Y to be the scaled score of a randomly selected student from this class. Find the
mean and standard deviation of Y.
b. What is the probability that a randomly selected student has a scaled test score of at least 90?
Mean of the Sum or Difference of Random Variables:
𝐸(𝑋 + 𝑌) = ____________
𝐸(𝑋 − 𝑌) = ____________
Note: In general, the mean of the sum of several
random variables is the sum of their means.
The mean of the difference of several random
variables is the difference of their means. The
order of subtraction is important!
Variance of the Sum and Difference of Independent Random Variables:
2
𝜎𝑋+𝑌
= ____________
2
𝜎𝑋−𝑌
= ____________
Note: In general, the variance of the sum or
difference of several independent random
variables is the sum of their variances.
6.3 Binomial and Geometric and Random Variables
A binomial setting arises when we perform several independent trials of the same chance process and
record the number of times that a particular outcome occurs.
The 4 conditions for a binomial setting are:
Binary?
Independent?
Number?
Success?
The probability distribution of X is a binomial distribution with parameters n and p, where

n is the ______________________________________________

p is the ______________________________________________
The possible values of X are the whole numbers from 0 to n.
Binomial coefficient: The number of ways of arranging k successes among n observations:
for k = 0, 1, 2, …, n where
𝑛! = 𝑛(𝑛 – 1)(𝑛 – 2) • … • (3)(2)(1)
and 0! = 1.
Binomial Probability:
If X has the binomial distribution with n trials and probability p of success on each trial, the possible values
of X are 0, 1, 2, …, n. If k is any one of these values,
𝑛
𝑃(𝑋 = 𝑘) = ( ) ∙ 𝑝𝑘 ∙ (1 − 𝑝)𝑛−𝑘
𝑘
Number of
arrangements
of k successes
Probability
of k
successes
Mean and Standard Deviation of a Binomial Random Variable:
𝜇𝑋 =
𝜎𝑋 =
Sampling without Replacement Condition:
Normal Approximation for Binomial Distributions:
We will use the Normal approximation when n is so larger that:
Probability of
n-k failures
A geometric setting arises when we perform independent trials of the same chance process and record
the number of trials until a particular outcome occurs. The four conditions for a geometric setting are:
Binary?
Independent?
Trials?
Success?
Geometric random variable: The number of trials Y that it takes to get a success in a geometric setting. The
probability distribution of Y is a geometric distribution with parameter p, the probability of a success on
any trial. The possible values of Y are 1, 2, 3, ….
Geometric Probability:
Mean (Expected Value) of a Geometric Random Variable