Discrete
Probability
Distributions
1
Probability
Basics
2
Probability Basics
• Random Experiment
• Sample Space, Ω
• Event
• Intersection, 𝐴 ∩ 𝐵
• Mutually exclusive, 𝐴 ∩ 𝐵 = ∅
• Union, 𝐴 ∪ 𝐵
• Complement, 𝐴𝐶
3
Probability Basics
Probability Postulates
1. If 𝐴 is any event in the sample space,Ω,
0≤ℙ 𝐴 ≤1
2. Let 𝐴 ⊆ Ω and let 𝑂𝑖 denote the basic outcomes. Then
ℙ 𝐴 = ෍ ℙ 𝑂𝑖
𝐴
3. ℙ Ω = 1.
4
Probability Basics
• Complement Rule: ℙ 𝐴𝑐 = 1 − ℙ 𝐴 .
• Addition Rule: ℙ 𝐴 ∪ 𝐵 = ℙ 𝐴 + ℙ 𝐵 − ℙ 𝐴 ∩ 𝐵
• Conditional Probability: ℙ 𝐴|𝐵
ℙ 𝐴∩𝐵
= ℙ𝐵 ;
ℙ 𝐵 ≠0
• The Multiplication Rule: ℙ 𝐴 ∩ 𝐵 = ℙ 𝐴|𝐵 ℙ 𝐵
5
Probability Basics
Two events, 𝐴 be and 𝐵,c are said to be statistically
independent if and only if
ℙ 𝐴∩𝐵 =ℙ 𝐴 ℙ 𝐵 .
6
Probability Basics
Bayes’ Theorem
Let 𝐴 and 𝐵 be two events. Then Bayes’ theorem states
that
ℙ 𝐴|𝐵 ℙ 𝐵
ℙ 𝐵|𝐴 =
.
ℙ 𝐴
7
Discrete Random
Variables
Probability Distributions
8
Random Variables
There are different types of variables we will consider:
9
Random Variables
Random Variables
• Discrete Random Variable
• Continuous Random Variables
Probability Distribution Function (pdf)
1. 0 ≤ ℙ 𝑋 = 𝑥 ≤ 1 ∀ x ∈ 𝑆𝑋
2. σ𝑥∈𝑆𝑋 ℙ 𝑋 = 𝑥 = 1
10
Discrete Random Variables
Cumulative Probability Distribution (cdf)
𝐹𝑋 𝑥0 = ෍ ℙ 𝑋 = 𝑥
𝑥≤𝑥0
1. 0 ≤ 𝐹𝑋 𝑥0 ≤ 1 ∀ 𝑥0 ∈ ℝ.
2. If 𝑥0 < 𝑥1 , then 𝐹𝑋 𝑥0 ≤ 𝐹𝑋 𝑥1 .
11
Discrete Random Variables
Expected Value
𝔼 𝑋 = 𝜇 = ෍ 𝑥ℙ 𝑋 = 𝑥
𝑥∈𝑆𝑥
Variance
𝜎2
= 𝔼 𝑋 − 𝜇 2 = 𝔼 𝑋 2 − 𝜇2
=
෍ 𝑥 2 ℙ 𝑋 = 𝑥 − 𝜇2
𝑥∈𝑆𝑋
The standard deviation, 𝜎 = 𝜎 2
12
Discrete Random Variables
13
Discrete Random Variables
Let 𝑋 be a random variable with mean 𝜇𝑋 and variance 𝜎𝑋2 , and
let 𝑎 and 𝑏 be any constant fixed numbers.
Define the random variable 𝑌 = 𝑎 + 𝑏𝑋. Then, the mean and
variance of Y are
𝜇𝑌
𝜎𝑌2
= 𝑎 + 𝑏𝜇𝑋
= 𝑏2 𝜎𝑋2
14
Discrete Random Variables
15
Multivariate Basics
Let 𝑋1 , … , 𝑋𝑛 be random variables,
• 𝔼 𝑋1 + ⋯ + 𝑋𝑛 = 𝔼 𝑋1 + ⋯ + 𝔼 𝑋𝑛
• 𝕍 𝑋𝑖 + 𝑋𝑗 = 𝕍 𝑋𝑖 + 𝕍 𝑋𝑗 + 2Cov 𝑋𝑖 , 𝑋𝑗
• 𝕍 σ𝑛𝑖=1 𝑋𝑖 = σ𝑛𝑖=1 𝕍 𝑋𝑖 + 2 σ1≤𝑖<𝑗≤𝑛 Cov 𝑋𝑖 , 𝑋𝑗
• Cov 𝑋𝑖 , 𝑋𝑖 = 𝕍 𝑋𝑖
16
Multivariate Basics
Double conditional expectation
Let 𝑋 and 𝑌 be two random variables. Then
𝔼 𝔼 𝑋|𝑌
=𝔼 𝑋
17
Binomial
Distribution
18
Binomial Distribution
• Two possible mutually exclusive and collectively exhaustive
outcomes.
• Let 𝑝 denote the probability of success, and the probability of
failure (1 − 𝑝).
1 if success
• 𝑋=ቊ
0 otherwise
This distribution is known as the Bernoulli distribution.
19
Binomial Distribution
Binomial distribution:
• Two possible outcomes for each trial.
• Mutually exclusive.
• Counting successes.
• Independent trials.
• Identical probability of success for all trials.
20
Binomial Distribution
Let 𝑋 ∼ Bin 𝑛, 𝑝 , then
ℙ 𝑋=𝑥
𝔼 𝑋
𝜎2
=
=
=
𝜎
=
𝑥
𝑛−𝑥
𝐶
𝑝
1
−
𝑝
𝑛 𝑥
𝑛𝑝
𝑛𝑝 1 − 𝑝
𝜎2
21
Binomial Distribution
22
Binomial Distribution
23
Binomial Distribution
24
Exercises
For each of the following, indicate if a discrete or a continuous
random variable provides the best definition:
a) The amount of oil exported by Saudi Arabia in January 2019.
b) The number of newspapers published by The Copenhagen
Post.
c) The number of rainy days in July at a beach resort.
d) The level of pressure in the tires of an automobile.
25
Exercises
• A presidential election poll contacts 2,000 randomly selected
people. Should the number of people that support candidate A
be analyzed using discrete or continuous probability models?
• The Embassy of Vietnam in India receives a number of visa
applications. Should the number of visas issued per day be
analyzed using discrete or continuous probability models?
26
Exercises
Show the probability distribution function of the number of
heads when three fair coins are tossed independently.
27
Exercises
The number of computers sold per day at Dan’s Computer Works
is defined by the following probability distribution:
𝒙
0
1
2
3
4
5
6
ℙ 𝑋=𝑥
0.03
0.11
0.15
0.22
0.19
0.26
0.04
a) ℙ 3 ≤ 𝑥 < 6 = ?
b) ℙ(𝑥 > 3) = ?
c) 𝑃(𝑥 ≤ 4) = ?
d) 𝑃(2 < 𝑥 ≤ 5) = ?
28
Exercises
Given the probability distribution function:
𝒙
0
1
2
ℙ 𝑋=𝑥
0.25
0.50
0.25
a) Graph the probability distribution function.
b) Calculate and graph the cumulative probability distribution.
c) Find the mean of the random variable 𝑋.
d) Find the variance of 𝑋.
29
Exercises
An automobile dealer calculates the proportion of new cars sold that
have been returned a various numbers of times for the correction of
defects during the warranty period. The results are shown in the
following table.
𝒙
0
1
2
3
4
ℙ 𝑋=𝑥
0.28
0.36
0.23
0.09
0.04
a) Graph the probability distribution function.
b) Calculate and graph the cumulative probability distribution.
c) Find the mean number of returns of an automobile for corrections
for defects during the warranty period.
d) Find the variance of the number of returns of an automobile for
corrections for defects during the warranty period.
30
Exercises
A production manager knows that 5% of components produced by a
particular manufacturing process have some defect. Six of these
components, whose characteristics can be assumed to be
independent of each other, are examined.
a) What is the probability that none of these components has a
defect?
b) What is the probability that one of these components has a
defect?
c) What is the probability that at least two of these components
have a defect?
31
Exercises
A local car dealer in Germany is mounting a new cross-divisional
promotional campaign. Purchasers of new cars may, if dissatisfied for
any reason, return the vehicle within 2 days of purchase and receive a
full refund. The cost to the dealer of such a refund is €300. The dealer
estimates that 18% of all purchasers will indeed return cars
purchased and obtain refunds. Suppose that 80 cars are purchased
during the campaign period.
a) Find the mean and standard deviation of the number of these cars
that will be returned for refunds.
b) Find the mean and standard deviation of the total refund costs
that will accrue as a result of these 80 purchases.
32