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)
(1,2)~(2,1)
(1,3)~(2,2)~(3,1)
(1,4)~(2,3)~(3,2)~(4,1)
(1,5)~(2,4)~(3,3)~(4,2)~(5,1)
(1,6)~(2,5)~(3,4)~(4,3)~(5,2)~(6,1)
(2,6)~(3,5)~(4,4)~(5,3)~(6,2)
(3,6)~(4,5)~(5,4)~(6,3)
(4,6)~(5,5)~(6,4)
(5,6)~(6,5)
(6,6)
\Omega
\mathbb{R}

In board games we care about the sum and not the outcomes which lead to the sum

2
3
4
5
6
7
8
9
10
11
12
\mathbb{R}

Q of Interest: What is the probability that the sum will  be 10?

Mapping outcomes to

\Omega
\mathbb{R}

In board games we care about the sum and not the outcomes which lead to the sum

\mathbb{R}

Q of Interest: What is the probability that the sum will  be 10?

\Omega
4

Q of Interest: What is the probability that a student's CGPA is 4.5?

Mapping outcomes to

\mathbb{R}
4.25
4.5
5
\mathbb{R}
CGPA
Height
Weight
Vit. D3
Age
\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

\mathbb{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

\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

\mathbb{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

\mathbb{R}
X: \Omega
\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

\mathbb{R}
\Omega
\mathbb{R}

Multiple functions (random variables) are possible for the given domain (sample space)

students

(domain)

X_1: height
\mathbb{R}
\mathbb{R}
X_1: weight
X_1: CGPA

Notation

X (n_1, n_2)

: sum of the numbers on two dice (n1 + n2)

Y (student)

: height of the student

X
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

(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)~?
\forall x \in \{2,3,4,5,6,7,8,9,10,11,12\}
X: \Omega
\mathbb{R}
P(X=x)
[0,1]
\{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)
\{1, 2,3,4,5,6\}

For discrete RVs we can think of the distribution as a table

X: \Omega
\mathbb{R}
P(X=x)
[0,1]
x
1
2
3
4
5
6
\frac{1}{6}
\frac{1}{6}
\frac{1}{6}
\frac{1}{6}
\frac{1}{6}
\frac{1}{6}

Assigning probabilities

\{2,3,4,5,6,7,8,9,10,11,12\}
X: \Omega
\mathbb{R}
P(X=x)
[0,1]
(1,1)
(1,2)~(2,1)
(1,3)~(2,2)~(3,1)
(1,4)~(2,3)~(3,2)~(4,1)
(1,5)~(2,4)~(3,3)~(4,2)~(5,1)
(1,6)~(2,5)~(3,4)~(4,3)~(5,2)~(6,1)
(2,6)~(3,5)~(4,4)~(5,3)~(6,2)
(3,6)~(4,5)~(5,4)~(6,3)
(4,6)~(5,5)~(6,4)
(5,6)~(6,5)
(6,6)
\Omega
2
3
4
5
6
7
8
9
10
11
12
x
Event: X = x
P(X = x)
\frac{1}{36}
\frac{2}{36}
\frac{3}{36}
\frac{4}{36}
\frac{5}{36}
\frac{6}{36}
\frac{5}{36}
\frac{4}{36}
\frac{3}{36}
\frac{2}{36}
\frac{1}{36}

Assigning probabilities

\{2,3,4,5,6,7,8,9,10,11,12\}
X: \Omega
\mathbb{R}
P(X=x)
[0,1]
2
3
4
5
6
7
8
9
10
11
12
x
P(X = x)
\frac{1}{36}
\frac{2}{36}
\frac{3}{36}
\frac{4}{36}
\frac{5}{36}
\frac{6}{36}
\frac{5}{36}
\frac{4}{36}
\frac{3}{36}
\frac{2}{36}
\frac{1}{36}

Assigning probabilities

\{2,3,4,5,6,7,8,9,10,11,12\}
X: \Omega
\mathbb{R}
p_X(x) = P(X=x) = P({\omega \in \Omega: X(\omega) = x})
[0,1]

Key idea:

Think of the event corresponding to 

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)

Probability Mass Function

p_{X}(x)

Probability Mass Function (PMF)

Probability Distribution

Distribution

Properties of a PMF

p_X(x) \geq 0
p_X(x) = P(X = x) = P(\{\omega \in \Omega: X(\omega) = x\} ) \geq 0
\sum_{x \in \mathbb{R}_X} p_X(x) = 1
\mathbb{R}_X \subset \mathbb{R}

(the set of values that the RV can take)

(the support of the RV)

p_X(x) \geq 0
\sum_{x \in \mathbb{R}_X} p_X(x) = 1

Properties of a PMF

\sum_{x \in \mathbb{R}_X} p_X(x) = 1
\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 

\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)