Chapter 4_Part B
Lecture Presentation Slides
Macmillan Learning © 2017
4.3 Random Variables
 Random variable
 Discrete random variables
 Continuous random variables
 Normal distributions as probability
distributions
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Random Variables
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A probability model describes the possible outcomes of a chance
process and the likelihood that those outcomes will occur.
A numerical variable that describes the outcomes of a chance process
is called a random variable. The probability model for a random
variable is its probability distribution.
A random variable takes numerical values that describe the outcomes
of some chance process.
The probability distribution of a random variable gives its possible
values and their probabilities.
Example: Consider tossing a fair coin three times.
Define X = the number of heads obtained
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
Value
0
1
2
3
Probability
1/8
3/8
3/8
1/8
Discrete Random Variable 1
There are two main types of random variables: discrete and continuous.
If we can find a way to list all possible outcomes for a random variable
and assign probabilities to each one, we have a discrete random
variable.
A discrete random variable X takes a fixed set of possible values with
gaps between. The probability distribution of a discrete random variable X
lists the values xi and their probabilities pi:
Value:
x1
Probability: p1
x2
p2
x3
p3
…
…
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. The sum of the probabilities is 1.
To find the probability of any event, add the probabilities pi of the particular
values xi that make up the event.
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Discrete Random Variable 2
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Example
A liberal arts college posts the grade distribution for its courses.
In a recent semester, students in one section of English 130
received 32% A’s, 42% B’s, 19% C’s, 3% D’s, and 4% F’s.
Here is the distribution of the discrete random variable called
“Grade Points,” where A = 4, B = 3, etc.:
Value:
0
Probability: 0.04
1
0.03
2
0.19
3
0.42
4
0.32
What is the probability that a randomly selected student got a B or
better?
P(X ≥ 3) = P(X = 3) + P(X = 4)
= 0.42 + 0.32 = 0.74.
Examples From Practice Sheet
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4.4 Means and Variances of
Random Variables
 The mean of a random variable
 Statistical estimation and the law of large
numbers
 Rules for means
 The variance of a random variable
 Rules for variances and standard deviations
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The Mean of a Random Variable
When analyzing discrete random variables, we will follow the same
strategy we used with quantitative data―describe the shape, center,
and spread and identify any outliers.
The mean of any discrete random variable is an average of the
possible outcomes, with each outcome weighted by its probability.
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Examples From Practice Sheet
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Example: Babies’ Health at Birth
The probability distribution for X = Apgar scores is shown below:
a. Show that the probability distribution for X is legitimate.
b. Make a histogram of the probability distribution. Describe what you see.
c. Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)?
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
(a) All probabilities are
between 0 and 1, and
they add up to 1. This
is a legitimate
probability distribution.
(c) P(X ≥ 7) = .908.
We’d have a 91%
chance of randomly
choosing a healthy
baby.
(b) The left-skewed shape of the distribution suggests a randomly selected
newborn will have an Apgar score at the high end of the scale. There is a small
chance of getting a baby with a score of 5 or lower.
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Example: Apgar Scores―What’s
Typical?
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Consider the random variable X = Apgar Score.
Compute the mean of the random variable X and interpret it in
context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
The mean Apgar score of a randomly selected newborn is 8.128. This is the
long-term average Apgar score of many, many randomly chosen babies.
Note: The expected value does not need to be a possible value of X or an
integer! It is a long-term average over many repetitions.
Statistical Estimation
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Suppose we would like to estimate an unknown µ. We could select an
SRS and base our estimate on the sample mean. However, a different
SRS would probably yield a different sample mean.
This basic fact is called sampling variability: The value of a statistic
varies in repeated random sampling.
To make sense of sampling variability, we ask, “What would happen if we
took many samples?”
Population
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
The Law of Large Numbers
How can x be an accurate estimate of μ? After all, different
random samples would produce different values of x.
If we keep on taking larger and larger samples, the statistic
x is guaranteed to get closer and closer to the parameter .
Draw independent observations at random from any population with
finite mean µ. The law of large numbers says that, as the number
of observations drawn increases, the sample mean of the observed
values gets closer and closer to the mean µ of the population.
Many people incorrectly believe the law of small numbers, which says that
we expect even short sequences of random events to show the kind of
average behavior that, in fact, appears only in the long run.
Our intuition does not do a good job of distinguishing random
behavior from systematic influences. This is why we need statistical
inference procedures, so we can verify that what we see in the data
is more than a random pattern.
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Rules for Means
Rules for Means
Rule 1: If X is a random variable and a and b are fixed numbers,
then
µa+bX = a + bµX.
Rule 2: If X and Y are random variables, then
µX+Y = µX + µY.
Rule 3: If X and Y are random variables, then
µX-Y = µX - µY.
Example: The crickets living in a field have a mean length of 1.2 inches.
What is the mean in centimeters? There are 2.54 cm in an inch, so the
mean length in inches is multiplied by 2.54: 1.2 x 2.54 = 3.05 cm. (Note
that we used Rule 1 with b = 1.2. There was no a for this situation.)
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Examples From Practice Sheet
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Variance of a Random Variable
Because we use the mean as the measure of center for a discrete
random variable, we’ll use the standard deviation as our measure of
spread. The definition of the variance of a random variable is similar to
the definition of the variance for a set of quantitative data.
To get the standard deviation of a random variable, take the square
root of the variance.
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Example: Apgar Scores―How
Variable Are They?
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Consider the random variable X = Apgar Score
Compute the standard deviation of the random variable X and interpret it in
context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
Variance
The standard deviation of X is 1.437. On average, a randomly selected
baby’s Apgar score will differ from the mean 8.128 by about 1.4 units.
Rules for Variances
Rules for Variances and Standard Deviations
Rule 1: If X is a random variable and a and b are fixed numbers,
then
σ2a+bX = b2σ2X.
Rule 2: If X and Y are independent random variables, then
σ2X+Y = σ2X + σ2Y,
σ2X-Y = σ2X + σ2Y.
Rule 3: If X and Y have correlation ρ, then
σ2X+Y = σ2X + σ2Y + 2ρσXσY ,
σ2X-Y = σ2X + σ2Y ̶ 2ρσXσY .
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Examples From Practice Sheet
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Conditional Probability
The probability we assign to an event can change if we know that some
other event has occurred. This idea is the key to many applications of
probability.
When we are trying to find the probability that one event will happen
under the condition that some other event is already known to have
occurred, we are trying to determine a conditional probability.
The probability that one event happens given that another event is
already known to have happened is called a conditional probability.
When P(A) > 0, the probability that event B happens given that event A
has happened is found by
P( B | A) 
P( A and B)
P( A)
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The General Multiplication Rule
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The definition of conditional probability reminds us that, in principle, all
probabilities, including conditional probabilities, can be found from the
assignment of probabilities to events that describe a random phenomenon.
The definition of conditional probability then turns into a rule for finding the
probability that both of two events occur.
The probability that events A and B both occur can be found using the general
multiplication rule:
P(A and B) = P(A) • P(B | A)
where P(B | A) is the conditional probability that event B occurs given that
event A has already occurred.
Examples From Practice Sheet
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Tree Diagrams
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Probability problems often require us to combine several of the basic rules
into a more elaborate calculation. One way to model chance behavior that
involves a sequence of outcomes is to construct a tree diagram.
Consider flipping a
coin twice.
What is the probability
of getting two heads?
Sample Space
HH HT TH TT
So, P(two heads) = P(HH) = 1/4
Tree Diagrams Example
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The Pew Internet and American Life Project finds that 93% of
teenagers (ages 12 to 17) use the Internet and that 55% of online teens
have posted a profile on a social-networking site.
What percent of teens are online and have posted a profile?
P(online )  0.93
P(profile | online )  0.55
P(online and have profile )  P(online )  P(profile | online )
 (0.93)(0.55)
 0.5115
51.15% of teens are online and have
posted a profile.

Bayes’s Rule

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Bayes’s Rule Example
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If a woman in her 20s gets screened for breast cancer and receives a positive test
result, what is the probability that she does have breast cancer?
Disease
incidence
Diagnosis
sensitivity 0.8
Positive
Cancer
0.0004
0.2
Mammography
0.1
A2 Positive
0.9996
No cancer
Incidence of breast
cancer among
women ages 20–30
Negative
False negative
False positive
A1
0.9
Diagnosis
specificity
Negative
Mammography
performance
P (pos | cancer ) P (cancer )
P (pos | cancer ) P (cancer )  P (pos | cancer ) P (no cancer )
0.8(0.0004)

 0 .3 %
0.8(0.004)  0.1(0.9996)
P (cancer | pos) 
Independence again
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If two events A and B do not influence each other, and if knowledge about
one does not change the probability of the other, the events are said to
be independent of each other.
Multiplication Rule for Independent Events
Two events A and B are independent if knowing that one occurs does
not change the probability that the other occurs. If A and B are
independent:
P(A and B) = P(A)  P(B)
Note: Two events A and B that both have positive probability are also
independent if:
P(B|A) = P(B)