# CS6015: Linear Algebra and Random Processes

## Lecture 30:  Random Variables, Types of Random Variables (discrete and continuous),  Probability Mass Function (PMF), Properties of PMF

### 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}

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

\Omega
\mathbb{R}

\mathbb{R}

\Omega
4

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

\mathbb{R}
\mathbb{R}

\Omega

\mathbb{R}
\mathbb{R}

\mathbb{R}
X: \Omega
\mathbb{R}

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

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

X (n_1, n_2)

Y (student)

X
Y

X

### 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\}

P(X = x)
\{1, 2,3,4,5,6\}

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]

X = x

P(X=x)

p_{X}(x)

### 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 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)

\Omega

\therefore