Chapter 5: Random Variables and
Discrete Probability Distributions
http://www.landers.co.uk/statistics-cartoons/
1
5.1-5.2: Random Variables - Goals
• Be able to define what a random variable is.
• Be able to differentiate between discrete and
continuous random variables.
• Describe the probability distribution of a discrete
random variable.
• Use the distribution and properties of a discrete
random variable to calculate the probability of an
event.
2
Random Variables
A random variable is a function that assigns a
unique numerical value to each outcome in a
sample space.
The rule for a random variable may be given by
a formula, a table, or words.
Random variables can either be discrete or
continuous.
3
Probability Distribution of a Random
Variables
• The probability distribution of a random
variable gives all of its possible values and the
probabilities for each of them.
4
Probability Distribution of a Random
Variables
• Probability mass function (pmf) is the
probability that a discrete random
variable is equal to some specific
value.
In symbols, p(x) = P(X = x)
Outcome
x1
probability p1
x2
p2
…
…
5
Examples: Probability Histograms
#1
0.4
0.2
0
1
2
3
4
Outcomes
Probability
Probability
0.6
0.6
#1
0.4
0.2
0
0
1
2
3
Outcomes
4
5
6
Examples: Probability Histograms
Probability
0.6
#2
0.4
0.2
0
Probability
1
2 Outcomes 3
0.6
4
#2
0.4
0.2
0
0
1
2
3
4
5
Outcomes
7
Properties of a Valid Probability
Distribution
1. 0 ≤ pi ≤ 1
2.
𝑝𝑖 (𝑥) = 1
𝑖
8
Example: Discrete Random Variable
In a standard deck of cards, we want to know
the probability of drawing a certain number of
spades when we draw 3 cards with replacement.
Let X be the number of spades that we draw.
a) What is the distribution?
b) Is this a valid distribution?
c) What is the probability that you draw at least
1 spade?
d) What is the probability that you draw at least
2 spades?
9
Probability
Example: Discrete (cont.)
0.6
0.5
0.4
0.3
0.2
0.1
0
Spades Example
0
1
2
3
Number of Spades
10
Example: Discrete Random Variable
In a standard deck of cards, we want to know
the probability of drawing a certain number of
spades when we draw 3 cards. Let X be the
number of spades that we draw.
a) What is the distribution?
b) Is this a valid distribution?
c) What is the probability that you draw at least
1 spade?
d) What is the probability that you draw at least
2 spades?
11
5.3: Mean, Variance, and Standard
Deviation for a Discrete Variable - Goals
• Be able to use a probability distribution to find the
mean of a discrete random variable.
• Calculate means using the rules for means (not in
the book)
• Be able to use a probability distribution to find the
variance and standard deviation of a discrete
random variable.
• Calculate variances (standard deviations) using the
rules for variances for both correlated and
uncorrelated random variables (not in the book)
12
Formula for the Mean of a Random
Variable
𝐸 𝑋 = 𝜇 = 𝜇𝑋
𝐸 𝑋 = 𝜇 = 𝜇𝑋 =
𝑥𝑖 𝑝𝑖
𝑖
13
Example: Expected value
What is the expected value of the following:
a) A fair 4-sided die
X
1
2
3
4
Probability 0.25 0.25 0.25 0.25
14
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 is a random variable and g is a function
of X, then:
𝐸 𝑔 𝑋
=
𝑔(𝑥𝑖 )𝑝𝑖
15
Example: Expected Value
An individual who has automobile insurance form a
certain company is randomly selected. Let X be the
number of moving violations for which the individual
was cited during the last 3 years. The distribution of X
is
X
0
1
2
3
px
0.60
0.25
0.10
0.05
a) Verify that E(X) = 0.60.
b) If the cost of insurance depends on the following
function of accidents, g(x) = 400 + (100x -15), what
is the expected value of the cost of the insurance?
16
Example: Expected Value
Five individuals who have automobile insurance from a
certain company are randomly selected. Let X and Y be
two different accident profiles in this insurance company:
X
px
0
0.60
1
0.25
2
0.10
3
0.05
Y
pY
0
0.40
1
0.35
2
0.15
3
0.10
E(X) = 0.60
E(Y) = 0.95
c) What is the expected value the total number of
accidents of the people if 2 of them have the
distribution in X and 3 have the distribution in Y?
17
Example: Expected value
An individual who has automobile insurance
form a certain company is randomly selected.
Let X be the number of moving violations for
which the individual was cited during the last 3
years. The distribution of X is
X
px
0
0.60
1
0.25
2
0.10
3
0.05
d) Calculate E(X2).
18
Variance of a Random Variable
𝑛
𝑖=1
2
𝑥
−
𝑥
𝑖
2
𝑠 =
𝑛−1
Var X = 𝜎 2 = 𝜎𝑋2
Var(X) = E X − 𝜇𝑋
2
=
(𝑥𝑖 − X )2 ∙ 𝑝𝑖
= E(X2) – (E(X))2
𝜎𝑋 =
𝑉𝑎𝑟(𝑋)
19
Example: Variance
An individual who has automobile insurance
form a certain company is randomly selected.
Let X be the number of moving violations for
which the individual was cited during the last 3
years. The distribution of X is
X
px
0
0.60
1
0.25
2
0.10
3
0.05
e) Calculate Var(X).
20
Rules for Variance
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
Rule 3: If X and Y have correlation ρ, then:
σ2XY = σ2X + σ2Y  2ρσXσY
21
Example: Variance
An individual who has automobile insurance form a
certain company is randomly selected. Let X be the
number of moving violations for which the individual was
cited during the last 3 years. The distribution of X is
X
px
0
0.60
1
0.25
2
0.10
3
0.05
a) Calculate the variance of this distribution.
b) If the cost of insurance depends on the following
function of accidents, g(x) = 400 + (100x -15), what is
the standard deviation of the cost of the insurance?
22
Example: Variance
5 individuals who have automobile insurance from a
certain company are randomly selected. Let X and Y
be two different independent accident profiles in this
insurance company:
X
0
1
2
3
px
0.60
0.25
0.10
0.05
Var(X) = 0.74
Y
pY
0
0.40
1
0.35
2
0.15
3
0.10
Var(Y) = 0.95
What is the standard deviation of the (2X – 3Y)?
23
5.4/5.5: Binomial and Poisson Distributions
- Goals
• Determine when the random variable X can be
modeled using the binomial or Poisson
Distributions.
• Calculate the probability, mean and standard
deviation when X has a binomial or Poisson
distribution.
24
Properties of a Binomial Experiment BInS
• Binary: There are only two possible outcomes
for each trial.
• Independent: The outcomes of the trials are
independent.
• n: The experiment consists of n identical trials
where n is fixed..
• Success: For each trial, the probability p of
success must be the same.
25
Binomial Setting: Example
Do the following use the Binomial Setting?
1. Rolling a fair 4-sided die five times and
observing whether the number showing is a 1
or not
2. In a drug trial, 20 patients with the same
condition are given a drug and some are given a
placebo to see if the drug is effective or not.
3. In quality control we want to see if a particular
product is ‘not acceptable’. We take 20 random
samples from an assembly line that uses
different machines to produce the product.
26
Binomial Distribution
The binomial random variable maps each
outcome in a binomial experiment to a real
number, and is defined to be the number of
successes in n trials.
• X ~ B(n,p)
27
Examples of Binomial Distribution
1. In a clinical trial, a patient’s condition may
improve or not. We study the number of
patients who improved.
2. Was a sales transaction considered pleasant?
The binomial distribution describes the
number of pleasant transactions.
3. In quality control we assess the number of
defective items in a lot of goods.
28
Binomial Probabilities
Suppose X is a binomial random variable with n
trials and probability of a success p. Then
𝑛 𝑥
𝑃 𝑋=𝑥 =
𝑝 (1 − 𝑝)𝑛−𝑥 , 𝑥 = 0,1,2, … , 𝑛
𝑥
𝑛
𝑛!
=
𝑥
𝑥! 𝑛 − 𝑥 !
29
Example: Binomial Distribution
Suppose 20% of all copies of a particular
textbook fail a certain binding strength test.
Let's check a batch of 15 such textbooks.
a) Is this a binomial distribution?
b) What is the chance that there are no
defective textbooks?
c) What is the chance that we get less than 3
defective textbooks?
d) What is the chance that we get more than 2
defective textbooks?
30
Example: Binomial Distribution (cont)
0.3
Proportion
0.25
0.2
n=15
p=0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
# of defective textbooks
31
Histograms of Binomial Distributions
0.2
0.15
0.1
0.2
0.15
0.1
0.05
0.05
0
0
0
1
2
3
4
5
6
7
8
9
10
Number of successes
0.3
0
1
2
3
4
5
6
7
8
9
10
Number of successes
n = 10
p = 0.75
0.25
P(X=x)
0.25
P(X=x)
n = 10
p = 0.25
0.25
P(X=x)
n = 10
p = 0.5
0.3
0.3
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
Number of successes
8
9
10
32
Cumulative Probabilities (CDF)
The Cumulative Probability Function is defined
as the following probability: P(X ≤ x).
33
Binomial Distribution: Mean and Standard
Deviation
If X ~ B(n,p) then
E(X) = X = np
𝜎𝑋 = 𝑛𝑝(1 − 𝑝)
34
Example: Binomial Distribution (cont)
Suppose 20% of all copies of a particular
textbook fail a certain binding strength test.
Let's check a batch of 15 such textbooks.
e) What are the mean and standard deviation of
the number of textbooks that will fail the
binding test?
35
Poisson Random Variable
• The Poisson random variable is a count of the
number of times the specific event occurs during
a given interval.
• Example:
– The number of people who enter the Union
from noon to 1 pm.
– The number of α-particles emitted from
Uranium-238 in 1 minute.
– The number of DNA fragments found from a
sequencing experiment.
– The number of dead trees in a square mile of
forest.
36
Poisson Experiment
1. The probability that a particular event will occur
in a given interval (of time, length, volume, etc.)
is the same for all units of equal size and is
proportional to the size of the unit.
2. The number of events that occur in any interval
is independent of the number that occur in any
other non-overlapping interval.
3. The probability that more than one event occurs
in a unit of measure is negligible for very smallsized units.
37
Poisson Distribution
X ~ Poisson()
𝑒 −𝜆 𝜆𝑥
𝑝 𝑥 =𝑃 𝑋=𝑥 =
, 𝑥 = 0, 1, 2, …
𝑥!
𝜆>0
X = 2 = 
𝜎𝑋 = 𝜆
38
Example: Poisson Distribution
An IT consultant receives an average of 3 calls per
hour. Let X be the number of calls the consultant
receives. Assume X follows a Poisson distribution.
a) What is the chance that the consultant receives
exactly one call during the next hour?
b) What is the chance that the consultant receives
more than one call during the next hour?
c) What is the chance that the consultant receives
exactly 5 calls during the next two hours?
39