CS6015: Linear Algebra and Random Processes
Lecture 30: Random Variables, Types of Random Variables (discrete and continuous), Probability Mass Function (PMF), Properties of PMF
Learning Objectives
What are random variables?
What is a probability mass function (PMF)?
What are the properties of a PMF?
Random Variable
Recap
Experiments, sample spaces, events
Axioms and Laws of Probability
This chapter: Focus on numerical quantities associated with the outcomes of experiments
Mapping outcomes to
(1,1)
(1,1)
(1,2) (2,1)
(1,2)~(2,1)
(1,3) (2,2) (3,1)
(1,3)~(2,2)~(3,1)
(1,4) (2,3) (3,2) (4,1)
(1,4)~(2,3)~(3,2)~(4,1)
(1,5) (2,4) (3,3) (4,2) (5,1)
(1,5)~(2,4)~(3,3)~(4,2)~(5,1)
(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
(1,6)~(2,5)~(3,4)~(4,3)~(5,2)~(6,1)
(2,6) (3,5) (4,4) (5,3) (6,2)
(2,6)~(3,5)~(4,4)~(5,3)~(6,2)
(3,6) (4,5) (5,4) (6,3)
(3,6)~(4,5)~(5,4)~(6,3)
(4,6) (5,5) (6,4)
(4,6)~(5,5)~(6,4)
(5,6) (6,5)
(5,6)~(6,5)
(6,6)
(6,6)
Ω
\Omega
R
\mathbb{R}
In board games we care about the sum and not the outcomes which lead to the sum
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
R
\mathbb{R}
Q of Interest: What is the probability that the sum will be 10?
Mapping outcomes to
Ω
\Omega
R
\mathbb{R}
In board games we care about the sum and not the outcomes which lead to the sum
R
\mathbb{R}
Q of Interest: What is the probability that the sum will be 10?
Ω
\Omega
4
4
Q of Interest: What is the probability that a student's CGPA is 4.5?
Mapping outcomes to
R
\mathbb{R}
4.25
4.25
4.5
4.5
5
5
R
\mathbb{R}
CGPA
CGPA
Height
Height
Weight
Weight
Vit.D3
Vit. D3
Age
Age
Ω
\Omega
Qs of Interest:
What is the probability that an employee has 2 children?
What is the probability that an employee's monthly salary is greater than 50K?
Mapping outcomes to
R
\mathbb{R}
R
\mathbb{R}
Experiment: Randomly select an employee
: All employees of the organisation
: Number of years of experience, number of projects, salary, income tax, num. children
Ω
\Omega
Qs of Interest:
What is the probability that the size of the farm is less than 2 acres?
What is the probability that the total yield is greater than 1 ton?
Mapping outcomes to
R
\mathbb{R}
R
\mathbb{R}
Experiment: Randomly select a farm
: All farms in the state
: size of the farm, total yield, soil moisture, water content
Mapping outcomes to
R
\mathbb{R}
X:Ω
X: \Omega
R
\mathbb{R}
A random variable is a function from a set of possible outcomes to the set of real numbers
(could be a subset of R)
function
(random variable)
domain
range
Mapping outcomes to
R
\mathbb{R}
Ω
\Omega
R
\mathbb{R}
Multiple functions (random variables) are possible for the given domain (sample space)
students
(domain)
X1:height
X_1: height
R
\mathbb{R}
R
\mathbb{R}
X1:weight
X_1: weight
X1:CGPA
X_1: CGPA
Notation
X(n1,n2)
X (n_1, n_2)
: sum of the numbers on two dice (n1 + n2)
Y(student)
Y (student)
: height of the student
X
X
Y
Y
Unlike functions we don't write brackets and arguments!
Questions
What are the values that the random variable can take?
What are the probabilities of the values that the random variable can take?
X
X
(discrete or continuous)
(we will return back to this later)
Types of random variables
Discrete
(finite or countably infinite)
the sum of the numbers on two dice
the outcome of a single die
the number of tosses after which a heads appears (countably infinite)
the number of children that an employee has
the number of cars in an image
Types of random variables
Continuous
the amount of rainfall in Chennai
the temperature of a surface
the density of a liquid
the height of a student
the haemoglobin level of a patient
Probability Mass Function
What are the probabilities of the values that a discrete random variable can take?
Assigning probabilities
Q of Interest: What is the probability that the value of the random variable will be x?
P(X=x) ?
P(X = x)~?
∀x∈{2,3,4,5,6,7,8,9,10,11,12}
\forall x \in \{2,3,4,5,6,7,8,9,10,11,12\}
X:Ω
X: \Omega
R
\mathbb{R}
P(X=x)
P(X=x)
[0,1]
[0,1]
{2,3,4,5,6,7,8,9,10,11,12}
\{2,3,4,5,6,7,8,9,10,11,12\}
An assignment of probabilities to all possible values that a discrete RV can take is called the distribution of the discrete random variable
Assigning probabilities
P(X=x)
P(X = x)
{1,2,3,4,5,6}
\{1, 2,3,4,5,6\}
For discrete RVs we can think of the distribution as a table
X:Ω
X: \Omega
R
\mathbb{R}
P(X=x)
P(X=x)
[0,1]
[0,1]
x
x
1
1
2
2
3
3
4
4
5
5
6
6
61
\frac{1}{6}
61
\frac{1}{6}
61
\frac{1}{6}
61
\frac{1}{6}
61
\frac{1}{6}
61
\frac{1}{6}
Assigning probabilities
{2,3,4,5,6,7,8,9,10,11,12}
\{2,3,4,5,6,7,8,9,10,11,12\}
X:Ω
X: \Omega
R
\mathbb{R}
P(X=x)
P(X=x)
[0,1]
[0,1]
(1,1)
(1,1)
(1,2) (2,1)
(1,2)~(2,1)
(1,3) (2,2) (3,1)
(1,3)~(2,2)~(3,1)
(1,4) (2,3) (3,2) (4,1)
(1,4)~(2,3)~(3,2)~(4,1)
(1,5) (2,4) (3,3) (4,2) (5,1)
(1,5)~(2,4)~(3,3)~(4,2)~(5,1)
(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
(1,6)~(2,5)~(3,4)~(4,3)~(5,2)~(6,1)
(2,6) (3,5) (4,4) (5,3) (6,2)
(2,6)~(3,5)~(4,4)~(5,3)~(6,2)
(3,6) (4,5) (5,4) (6,3)
(3,6)~(4,5)~(5,4)~(6,3)
(4,6) (5,5) (6,4)
(4,6)~(5,5)~(6,4)
(5,6) (6,5)
(5,6)~(6,5)
(6,6)
(6,6)
Ω
\Omega
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
x
x
Event:X=x
Event: X = x
P(X=x)
P(X = x)
361
\frac{1}{36}
362
\frac{2}{36}
363
\frac{3}{36}
364
\frac{4}{36}
365
\frac{5}{36}
366
\frac{6}{36}
365
\frac{5}{36}
364
\frac{4}{36}
363
\frac{3}{36}
362
\frac{2}{36}
361
\frac{1}{36}
Assigning probabilities
{2,3,4,5,6,7,8,9,10,11,12}
\{2,3,4,5,6,7,8,9,10,11,12\}
X:Ω
X: \Omega
R
\mathbb{R}
P(X=x)
P(X=x)
[0,1]
[0,1]
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
x
x
P(X=x)
P(X = x)
361
\frac{1}{36}
362
\frac{2}{36}
363
\frac{3}{36}
364
\frac{4}{36}
365
\frac{5}{36}
366
\frac{6}{36}
365
\frac{5}{36}
364
\frac{4}{36}
363
\frac{3}{36}
362
\frac{2}{36}
361
\frac{1}{36}
Assigning probabilities
{2,3,4,5,6,7,8,9,10,11,12}
\{2,3,4,5,6,7,8,9,10,11,12\}
X:Ω
X: \Omega
R
\mathbb{R}
pX(x)=P(X=x)=P(ω∈Ω:X(ω)=x)
p_X(x) = P(X=x) = P({\omega \in \Omega: X(\omega) = x})
[0,1]
[0,1]
Key idea:
Think of the event corresponding to
X=x
X = x
Once we know this event (subset of sample space) we know how to compute P(X=x)
(Probability distribution of the random variable X)
P(X=x)
P(X=x)
Probability Mass Function
pX(x)
p_{X}(x)
Probability Mass Function (PMF)
Probability Distribution
Distribution
Properties of a PMF
pX(x)≥0
p_X(x) \geq 0
pX(x)=P(X=x)=P({ω∈Ω:X(ω)=x})≥0
p_X(x) = P(X = x) = P(\{\omega \in \Omega: X(\omega) = x\} ) \geq 0
∑x∈RXpX(x)=1
\sum_{x \in \mathbb{R}_X} p_X(x) = 1
RX⊂R
\mathbb{R}_X \subset \mathbb{R}
(the set of values that the RV can take)
(the support of the RV)
pX(x)≥0
p_X(x) \geq 0
∑x∈RXpX(x)=1
\sum_{x \in \mathbb{R}_X} p_X(x) = 1
Properties of a PMF
∑x∈RXpX(x)=1
\sum_{x \in \mathbb{R}_X} p_X(x) = 1
∑x∈RXpX(x)=∑x∈RXP(X=x)
\sum_{x\in \mathbb{R}_X} p_X(x) = \sum_{x\in \mathbb{R}_X} P(X =x)
RHS is the sum of the probabilities of disjoint events which partition
Ω
\Omega
RHS sums to 1
∴
\therefore
Proof:
Learning Objectives
What are random variables?
What is a probability mass function (PMF)?
What are the properties of a PMF?
(achieved)
CS6015: Linear Algebra and Random Processes Lecture 30: Random Variables, Types of Random Variables (discrete and continuous), Probability Mass Function (PMF), Properties of PMF
CS6015: Lecture 30
By Mitesh Khapra
CS6015: Lecture 30
Lecture 30:Random Variables, Types of Random Variables (discrete and continuous), Probability Mass Function (PMF), Properties of PMF
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