Cumulative Probability Mass Function of a Discrete Random Variable

Probability Function

A probability functions is a function that gives the relationship between the variable in a probability distribution and its probability. It gives the probability for the variable to assume a certain value.

The probability function is denoted as f(x) = P(X=x)

Eg:	1.	The probability distribution of a random variable relating to the experiment of tossing 3 coins where getting a head is termed a success and "x" is a variable indicating the number of successes. x 0 1 2 3 P(X=x) 1 8 3 8 3 8 1 8 Where, f(x) = P(X=x) represents the probability function relating to this probability distribution Where "x =0", f(0) = P(X=0); Where "x = 2" f(2) = P(X=2); ...

Cumulative Probability Function

The cumulative probability function of a random variable (discrete or continuous) is a function whose domain is similar to that of the probability mass or density function, but whose range is the set of probabilities associated with the possibility that the random variable will assume a value that is less than or equal to the values in the domain. The cumulative probability function of a random variable X is denoted by F_X(x) and is defined as F_X(x) = P (X ≤ x)

Every probability distribution has a respective cumulative probability distribution. The cumulative probability distribution of a random variable relating to the experiment of tossing 3 coins where getting a head is termed a success and "x" is a variable indicating the number of successes.

x	0	1	2	3
f(x) {=P(X=x)}	1 8	3 8	3 8	1 8
FX(x) { P(X ≤ x)}	1 8	4 8	7 8	8 8

Where, F_X(x) = P(X=x) represents the cumulative probability function relating to this probability distribution

Where

"x =0", F(0) = P(X = 0);

"x = 1" F(1) = P(X ≤ 1) ⇒ F(2) = P(X = 0) + P(X = 1)

"x = 2" F(2) = P(X ≤ 2) ⇒ F(2) = P(X = 0) + P(X = 1) + P(X = 2)

"x = 3" F(2) = P(X ≤ 2) ⇒ F(2) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)