Discrete probability distribution

To fully understand the characteristics of a discrete probability distribution, we should know four parts of the distribution.

(1) Probability mass function (pmf)

(2) Cumulative distribution function (cdf)

(3) Average

(4) Variable transformation

Let's say that there is

[1] [1] [1] [1]

[2] [2] [2] [2] [2] [2]

What is the probability distribution of X?

X = the number of the card. randomly selected from the box

To get pmf...

1) Pick random variable

2) Domain X = 1, 2.

X	1	2
P(X)	0.4	0.6

cdf will be

P(X <= a ) 0 (a<1)

0.4 (1<= a < 2)

0.6 ( a >=2)

P(X <=a) = F(a)

When we wanna calculate the average,
we can use pdf by multiplying the X and P(X).

$E(X) = x_{i} * P(x_{i})$

It will be 1*0.4 + 2*0.6 = 1.6

Here, we can transform the variable from x to $x^{2}$ .
We should fill out the table with new variable $x^{2}$ .

g(x) = $x^{2}$	g(1)=1^2=1	g(2)=2^2= 4
P(X)	0.4	0.6

With transformed table, we can calculate the average again.

$E(g(x)) = \sum_{i} g(x_{i})*P(x_{i})$

So it will be 1*0.4 + 4*0.6 = 2.8