Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
1/23
Probability & Statistics
Discrete Random Variables 1
TRAN Hoang Tung
NGUYEN Minh Huong
NGUYEN Mai Chung
University of Science and Technology of Hanoi (USTH)
September 12, 2019

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence
Counting

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence
Counting
Today

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence
Counting
Today
Discrete random variable

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence
Counting
Today
Discrete random variable
Probability mass function (PMF)

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
2/23
The Course So Far
What we have learnt?
Probability model, conditional probability, total
probability theorem, Bayes’ rule
Independence, conditional independence
Counting
Today
Discrete random variable
Probability mass function (PMF)
Expectation: mean, variance

Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
3/23
1
Discrete Random Variables
Definition
Probability Mass Function
2
Common Random Variables
Bernoulli
Uniform
Binomial
Geometric
3
Expectation
The Mean
Expected Value Rule
The Variance

Probability &
Statistics
Discrete
Random
Variables
Definition
Probability
Mass Function
Common
Random
Variables
Expectation
4/23
1
Discrete Random Variables
Definition
Probability Mass Function
2
Common Random Variables
3
Expectation

Probability &
Statistics
Discrete
Random
Variables
Definition
Probability
Mass Function
Common
Random
Variables
Expectation
5/23
Discrete Random Variables
An example

Probability &
Statistics
Discrete
Random
Variables

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