Chapter 11
Section 1
Random Variables
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 1 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 2 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 3 of 34
Chapter 11 – Section 1
● A random variable is a numeric measure of the
outcome of a probability experiment
 Random variables reflect measurements that can
change as the experiment is repeated
 Random variables are denoted with capital letters,
typically using X (and Y and Z …)
 Values are usually written with lower case letters,
typically using x (and y and z ...)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 4 of 34
Chapter 11 – Section 1
● Examples
● Tossing
Tossing four
four coins
coins and
and counting
counting the
the number
number of
of
heads
 The number could be 0, 1, 2, 3, or 4
 The number could change when we toss another four
coins
● Measuring the heights of students
 The heights could change from student to student
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 5 of 34
Chapter 11 – Section 1
● A discrete random variable is a random variable
that has either a finite or a countable number of
values
 A finite number of values such as {0, 1, 2, 3, and 4}
 A countable number of values such as {1, 2, 3, …}
● Discrete random variables are designed to
model discrete variables
● Discrete random variables are often “counts of
…”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 6 of 34
Chapter 11 – Section 1
● An example of a discrete random variable
● The number of heads in tossing 3 coins (a finite
number of possible values)
 There are four possible values – 0 heads, 1 head, 2
heads, and 3 heads
 A finite number of possible values – a discrete
random variable
 This fits our general concept that discrete random
variables are often “counts of …”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 7 of 34
Chapter 11 – Section 1
● Other examples of discrete random variables
● The possible rolls when rolling a pair of dice
 A finite number of possible pairs, ranging from (1,1) to
(6,6)
● The number of pages in statistics textbooks
 A countable number of possible values
● The number of visitors to the White House in a
day
 A countable number of possible values
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 8 of 34
Chapter 11 – Section 1
● A continuous random variable is a random
variable that has an infinite, and more than
countable, number of values
 The values are any number in an interval
● Continuous random variables are designed to
model continuous variables (see section 1.1)
● Continuous random variables are often
“measurements of …”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 9 of 34
Chapter 11 – Section 1
● An example of a continuous random variable
● The possible temperature in Chicago at noon
tomorrow, measured in degrees Fahrenheit
 The possible values (assuming that we can measure
temperature to great accuracy) are in an interval
 The interval may be something like (–20,110)
 This fits our general concept that continuous random
variables are often “measurements of …”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 10 of 34
Chapter 11 – Section 1
● Other examples of continuous random variables
● The height of a college student
 A value in an interval between 3 and 8 feet
● The length of a country and western song
 A value in an interval between 1 and 15 minutes
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 11 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 12 of 34
Chapter 11 – Section 1
● The probability distribution of a discrete random
variable X relates the values of X with their
corresponding probabilities
● A distribution could be
 In the form of a table
 In the form of a graph
 In the form of a mathematical formula
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 13 of 34
Chapter 11 – Section 1
● If X is a discrete random variable and x is a
possible value for X, then we write P(x) as the
probability that X is equal to x
● Examples
 In tossing one coin, if X is the number of heads, then
P(0) = 0.5 and P(1) = 0.5
 In rolling one die, if X is the number rolled, then
P(1) = 1/6
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 14 of 34
Chapter 11 – Section 1
● Properties of P(x)
● Since P(x) form a probability distribution, they
must satisfy the rules of probability
 0 ≤ P(x) ≤ 1
 Σ P(x) = 1
● In the second rule, the Σ sign means to add up
the P(x)’s for all the possible x’s
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 15 of 34
Chapter 11 – Section 1
● An example of a discrete probability distribution
x
1
2
P(x)
.2
.6
5
6
.1
.1
● All of the P(x) values are positive and they add
up to 1
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 16 of 34
Chapter 11 – Section 1
● An example that is not a probability distribution
x
1
2
P(x)
.2
.6
5
6
-.3
.1
● Two things are wrong
 P(5) is negative
 The P(x)’s do not add up to 1
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 17 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 18 of 34
Chapter 11 – Section 1
● A probability histogram is a histogram where
 The horizontal axis corresponds to the possible
values of X (i.e. the x’s)
 The vertical axis corresponds to the probabilities for
those values (i.e. the P(x)’s)
● A probability histogram is very similar to a
relative frequency histogram
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 19 of 34
Chapter 11 – Section 1
● An example of a probability histogram
● The histogram is drawn so that the height of the
bar is the probability of that value
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 20 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 21 of 34
Chapter 11 – Section 1
● The mean of a probability distribution can be
thought of in this way:
 There are various possible values of a discrete
random variable
 The values that have the higher probabilities are the
ones that occur more often
 The values that occur more often should have a
larger role in calculating the mean
 The mean is the weighted average of the values,
weighted by the probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 22 of 34
Chapter 11 – Section 1
● The mean of a discrete random variable is
μX = Σ [ x • P(x) ]
● In this formula
 x are the possible values of X
 P(x) is the probability that x occurs
 Σ means to add up these terms for all the possible
values x
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 23 of 34
Chapter 11 – Section 1
● Example of a calculation for the mean
Multiply
x
P(x)
x • P(x)
1
0.2
0.2
2
0.6
1.2
Multiply again
5
0.1
0.5
Multiply again
6
0.1
0.6
Multiply again
● Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5
● The mean of this discrete random variable is 2.5
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 24 of 34
Chapter 11 – Section 1
● The calculation for this problem written out
μX = Σ [ x • P(x) ]
= [1• 0.2] + [2• 0.6] + [5• 0.1] + [6• 0.1]
= 0.2 + 1.2 + 0.5 + 0.6
= 2.5
● The mean of this discrete random variable is 2.5
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 25 of 34
Chapter 11 – Section 1
● The mean can also be thought of this way (as in
the Law of Large Numbers)
 If we repeat the experiment many times
 If we record the result each time
 If we calculate the mean of the results (this is just a
mean of a group of numbers)
 Then this mean of the results gets closer and closer
to the mean of the random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 26 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 27 of 34
Chapter 11 – Section 1
● The expected value of a random variable is
another term for its mean
● The term “expected value” illustrates the long
term nature of the experiments – as we perform
more and more experiments, the mean of the
results of those experiments gets closer to the
“expected value” of the random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 28 of 34
Chapter 11 – Section 1
● Learning objectives
1

2

3

4

5

6

Distinguish between discrete and continuous random
variables
Identify discrete probability distributions
Construct probability histograms
Compute and interpret the mean of a discrete random
variable
Interpret the mean of a discrete random variable as
an expected value
Compute the variance and standard deviation of a
discrete random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 29 of 34
Chapter 11 – Section 1
● The variance of a discrete random variable is
computed similarly as for the mean
● The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
● The variance is the weighted sum of the squared
differences from the mean
σX2 = Σ [ (x – μX)2 • P(x) ]
● The standard deviation, as we’ve seen before, is
the square root of the variance … σX = √ σX2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 30 of 34
Chapter 11 – Section 1
● The variance formula
σX2 = Σ [ (x – μX)2 • P(x) ]
can involve calculations with many decimals or
fractions
● An equivalent formula is
σX2 = [ Σ x2 • P(x) ] – μX2
● This formula is often easier to compute
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 31 of 34
Chapter 11 – Section 1
● For variables and samples, we had the concept
of a population variance (for the entire
population) and a sample variance (for a sample
from that population)
● These probability distributions model the
complete population
 These are population variance formulas
 There is no analogy for sample variance here
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 32 of 34
Chapter 11
Section 2
The Binomial
Probability Distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 33 of 34
Chapter 11 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 34 of 34
Chapter 11 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 35 of 34
Chapter 11 – Section 2
● A binomial experiment has the following
structure
 The first test is performed … the result is either a
success or a failure
 The second test is performed … the result is either a
success or a failure. This result is independent of the
first and the chance of success is the same
 A third test is performed … the result is either a
success or a failure. The result is independent of the
first two and the chance of success is the same
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 36 of 34
Chapter 11 – Section 2
● Example
 A card is drawn from a deck. A
A “success”
“success” is
is for
for that
that
card to be a heart … a “failure” is for any other suit
 The card is then put back into the deck
 A second card is drawn from the deck with the same
definition of success.
 The second card is put back into the deck
 We continue for 10 cards
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 37 of 34
Chapter 11 – Section 2
● A binomial experiment is an experiment with the
following characteristics
 The experiment is performed a fixed number of times,
each time called a trial
 The trials are independent
 Each trial has two possible outcomes, usually called a
success and a failure
 The probability of success is the same for every trial
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 38 of 34
Chapter 11 – Section 2
● Notation used for binomial distributions
 The number of trials is represented by n
 The probability of a success is represented by p
 The total number of successes in n trials is
represented by X
● Because there cannot be a negative number of
successes, and because there cannot be more
than n successes (out of n attempts)
0≤X≤n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 39 of 34
Chapter 11 – Section 2
● In our card drawing example
 Each trial is the experiment of drawing one card
 The experiment is performed 10 times, so n = 10
 The trials are independent because the drawn card is
put back into the deck
 Each trial has two possible outcomes, a “success” of
drawing a heart and a “failure” of drawing anything
else
 The probability of success is 0.25, the same for every
trial, so p = 0.25
 X, the number of successes, is between 0 and 10
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 40 of 34
Chapter 11 – Section 2
● The word “success” does not mean that this is a
good outcome or that we want this to be the
outcome
● A “success” in our card drawing experiment is to
draw a heart
● If we are counting hearts, then this is the
outcome that we are measuring
● There is no good or bad meaning to “success”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 41 of 34
Chapter 11 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 42 of 34
Chapter 11 – Section 2
● We would like to calculate the probabilities of X,
i.e. P(0), P(1), P(2), …, P(n)
● Do a simpler example first
 For n = 3 trials
 With p = .4 probability of success
 Calculate P(2), the probability of 2 successes
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 43 of 34
Chapter 11 – Section 2
● For 3 trials, the possible ways of getting exactly
2 successes are
 S S F
 S F S
 F S S
● The probabilities for each (using the
multiplication rule) are
 0.4 • 0.4 • 0.6 = 0.096
 0.4 • 0.6 • 0.4 = 0.096
 0.6 • 0.4 • 0.4 = 0.096
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 44 of 34
Chapter 11 – Section 2
● The total probability is
P(2) = 0.096 + 0.096 + 0.096 = 0.288
● But there is a pattern
 Each way had the same probability … the probability
of 2 success (0.4 times 0.4) times the probability of 1
failure (0.6)
● The probability for each case is
0.42 • 0.61
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 45 of 34
Chapter 11 – Section 2
● There are 3 cases
 S S F could represent choosing a combination of
2 out of 3 … choosing the first and the second
 S F S could represent choosing a second
combination of 2 out of 3 … choosing the first and the
third
 F S S could represent choosing a third
combination of 2 out of 3
● These are the 3 = 3C2 ways to choose 2 out of 3
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 46 of 34
Chapter 11 – Section 2
● Thus the total probability
P(2) = .096 + .096 + .096 = .288
can also be written as
P(2) = 3C2 • .42 • .61
● In other words, the probability is
 The number of ways of choosing 2 out of 3, times
 The probability of 2 successes, times
 The probability of 1 failure
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 47 of 34
Chapter 11 – Section 2
● The general formula for the binomial
probabilities is just this
● For P(x), the probability of x successes, the
probability is
 The number of ways of choosing x out of n, times
 The probability of x successes, times
 The probability of n-x failures
● This formula is
P(x) = nCx px (1 – p)n-x
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 48 of 34
Chapter 11 – Section 2
● Example
● A student guesses at random on a multiple
choice quiz
 There are n = 10 questions in total
 There are 5 choices per question so that the
probability of success p = 1/5 = .2
● What is the probability that the student gets 6
questions correct?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 49 of 34
Chapter 11 – Section 2
● Example continued
● This is a binomial experiment
 There are a finite number n = 10 of trials
 Each trial has two outcomes (a correct guess and an
incorrect guess)
 The probability of success is independent from trial to
trial (every one is a random guess)
 The probability of success p = .2 is the same for each
trial
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 50 of 34
Chapter 11 – Section 2
● Example continued
● The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x
= 6C10 .26 .84
= 210 • .000064 • .4096
= .005505
● This is less than a 1% chance
● In fact, the chance of getting 6 or more correct
(i.e. a passing score) is also less than 1%
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 51 of 34
Chapter 11 – Section 2
● Binomial calculations can be difficult because of
the large numbers (the nCx) times the small
numbers (the px and (1-p)n-x)
● It is possible to use tables to look up these
probabilities
● It is best to use a calculator routine or a software
program to compute these probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 52 of 34
Chapter 11 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 53 of 34
Chapter 11 – Section 2
● We would like to find the mean of a binomial
distribution
● Example
 There are 10 questions
 The probability of success is .20 on each one
 Then the expected number of successes would be
10 • .20 = 2
● The general formula
μX = n p
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 54 of 34
Chapter 11 – Section 2
● We would like to find the standard deviation and
variance of a binomial distribution
● This calculation is more difficult
● The standard deviation is
σX = √ n p (1 – p)
and the variance is
σX2 = n p (1 – p)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 55 of 34
Chapter 11 – Section 2
● For our random guessing on a quiz problem
 n = 10
 p = .2
 x=6
● Therefore
 The mean is np = 10 • .2 = 2
 The variance is np(1-p) = 10 • .2 • .8 = .16
 The standard deviation is √.16 = .4
● Remember the empirical rule? A passing grade
of 6 is 10 standard deviations from the mean …
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 56 of 34
Chapter 11 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 57 of 34
Chapter 11 – Section 2
● With the formula for the binomial probabilities
P(x), we can construct histograms for the
binomial distribution
● There are three different shapes for these
histograms
 When p < .5, the histogram is skewed right
 When p = .5, the histogram is symmetric
 When p > .5, the histogram is skewed left
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 58 of 34
Chapter 11 – Section 2
● For n = 10 and p = .2 (skewed right)
 Mean = 2
 Standard deviation = .4
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 59 of 34
Chapter 11 – Section 2
● For n = 10 and p = .5 (symmetric)
 Mean = 5
 Standard deviation = .5
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 60 of 34
Chapter 11 – Section 2
● For n = 10 and p = .8 (skewed left)
 Mean = 8
 Standard deviation = .4
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 61 of 34
Chapter 11 – Section 2
● Despite binomial distributions being skewed, the
histograms appear more and more bell shaped
as n gets larger
● This will be important!
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 62 of 34
Summary: Chapter 11 – Section 2
● Binomial random variables model a series of
independent trials, each of which can be a
success or a failure, each of which has the same
probability of success
● The binomial random variable has mean equal
to np and variance equal to np(1-p)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 1 – Slide 63 of 34