Random Variables and Its Probability Distributions

A random variable is a real valued function whose domain is the sample space of a random experiment. 
For example, let us consider the experiment of tossing a coin two times in succession. 
The sample space of the experiment is  S = {HH, HT, TH, TT}. 
If X denotes the number of heads obtained, then X is a random variable and for each outcome, its value is as given below :
X(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0. 
More than one random variables can be defined on the same sample space. For example, let Y denote the number of heads minus the number of tails for each outcome of the above sample space S. 
Then Y(HH) = 2, Y (HT) = 0, Y (TH) = 0, Y (TT) = – 2. 
Thus, X and Y are two different random variables defined on the same sample space S. 

In most of the random experiments , we were not on;ly interested in the particular outcome that occurs but rather in some number associated with that outcomes.
Example : 

(i) In tossing two dice, we may be interested in the sum of the numbers on the two dice. 

(ii) In tossing a coin 50 times, we may want the number of heads obtained.

(iii) In the experiment of taking out four articles (one after the other) at random from a lot of 20 articles in which 6 are defective, we want to know the number of defectives in the sample of four and not in the particular sequence of defective and nondefective articles.
In all of the above experiments, We have a rule which assigns to each outcome of the experiment a single real number.
This single real number may vary with different outcomes of the experiment.
Its value depends upon the outcome of a random experiments.
It is called random variable.
Random variable is denoted by X. 

Video link : https://youtu.be/M-Wz5wRhaNQ

Probability distribution of a random variable:
The experiment of selecting one family out of ten families `f_1,  f_2 ,..., f_10` in such a manner that each family  is equally likely to be selected. Let the families `f_1,  f_2, ... , f_10`  have 3, 4, 3, 2, 5, 4, 3, 6, 4, 5 members, respectively. 
Let us select a family and note down the number of members in the family denoting X. Clearly, X is a random variable defined as below : 
`X(f_1) = 3, X(f_2) = 4, X(f_3) = 3, X(f_4) = 2, X(f_5) = 5, X(f_6) = 4, X(f_7) = 3, X(f_8) = 6, X(f_9) = 4, X(f_10) = 5 `
Thus, X can take any value 2,3,4,5 or 6 depending upon which family is selected. Now,  X will take the value 2 when the family `f_4` is selected.  X can take the value 3 when any one of the families `f_1, f_3, f_7` is selected. 
Similarly,
X = 4,  when family `f_2`, `f_6` or `f_9` is selected, 
X = 5,  when family `f_5` or `f_10` is selected 
and
X = 6, when family `f_8`  is selected.
Since we had assumed that each family is equally likely to be selected, the probability that family `f_4` is selected is `1/10`.
Thus, the probability that X can take the value 2 is `1/10`. We write P(X = 2) = `1/10`
Also, the probability that any one of the families` f_1, f_3 "or" f_7` is selected is `P({f_1, f_3, f_7}) `= `3/10`
Thus, the probability that X can take the value 3 = `3/10`
We write  P(X = 3) = `3/10`
Similarly, we obtain P(X = 4) = `P({f_2, f_6, f_9}) = 3/10`
P(X = 5) = `P({f_5, f_10}) = 2/10`
and P(X = 6) = `P({f_8}) = 1/10`
The values of the random variable along with the corresponding probabilities is called the probability distribution of the random variable X

Defination :  The probability distribution of a random variable X is the system of numbers

where , `p_i > 0  sum_(i=1)^n  p_i = 1 , i = 1 , 2,..., n`
The real numbers `x_1, x_2,..., x_n` are the possible values of the random variable X and `p_i (i = 1,2,..., n)` is the probability of the random variable X taking the value `x_i` i.e., `P(X = x_i) = p_i`
Video link : https://youtu.be/HYKdhi0c-KM