(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

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}

\{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}

\{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}

\{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)

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}

p_X(x) \geq 0

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

\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