How is probability distribution of random variable construct, Mathematics

How is the probability distribution of a random variable constructed?
Usually, the past behavior of the variable is studied and the frequency distribution of the past data is formed. If the past behavior can be taken as a representative pattern of the future also, then the past frequency distribution can be used as a guide for predicting the future values of the random variable. If past behavior cannot help, then a subjective probability distribution of the likely future values is formed on the basis of expert opinion.

Example

The issue pattern of a component AB323 as observed by the storekeeper of XYZ Limited over the past one hundred working days has been summarized below.

Number Issued	No. of days

100	1
101	2
102	3
103	5
104	6
105	7
106	9
107	10
108	12
109	11
110	9
111	8
112	6
113	5
114	4
115	2

From the above table, in addition to observing that the issues have ranged between 100 units to 115 units, the storekeeper can also assess the chances of each quantity being issued by forming a probability distribution.

The probability distribution can be formed by dividing the individual values by 100 (the total number of days over which the observations have been made).

Number issued	Probability that the Random Variable will take on this value

100	0.01
101	0.02
102	0.03
103	0.05
104	0.06
105	0.07
106	0.09
107	0.10
108	0.12
109	0.11
110	0.09
111	0.08
112	0.06
113	0.05
114	0.04
115	0.02
Total	1.00

The probability distribution of a random variable provides a probability for each possible value and the probabilities must sum to 1.

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