Cumulative Probability

In probability theory the cumulative distribution function (CDF) or just distribution function, completely describes the probability distribution of a real valued random variable X. Cumulative distribution functions are used to specify the distribution of multivariate random variables.

For every real number x, the CDF of real valued random variable is X its given by
                              f(x)=  P[X <= x]

where the function right hand side is represents the probability of  the random variable X takes on the value less than or equal to x. The probability of  X is lies in the interval (a, b) is therefore F_X (b) − F_X (a).

Cumulative Probability Example:

Consider a coin flip experiment. If we flip a coin two times, we might ask that what is the probability that the coin flips would result in one or fewer heads? The answer would be a cumulative probability. It would be the probability when that the coin is flip results in zero heads plus the probability that the coin flip results in one head. Thus, the cumulative probability would equal:

P(X < 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75

The table below shows that the both of the probabilities and the cumulative probabilities associated with this experiment.

Number of heads              Probability       Cumulative Probability

0                                            0.25                                0.25

1                                             0.50                                0.75

2                                             0.25                                1.00

 Assume we have a random variable X. Cumulative probabilities that are provide for each value x, the probability of a result less than or equal to X, P[X <= x].

Example:

Here's the probability distribution and for a discrete random variable X
X                  f(x)
1                   0.1
2                   0.2
3                    0.4
4                    0.3
The cumulative distribution function tables and for each value x = 1, 2, 3, 4, the probability of a result less than or equal

For example:
* P[ X <= 1 ] = 0.1

* P[ X <= 2 ] = 0.1 + 0.2 = 0.3

* P[ X <= 3 ] = 0.1 + 0.2 + 0.4 = 0.7

* P[ X <= 4 ] = 0.1 + 0.2 + 0.4 + 0.3 = 1

 These probabilities can be tabled

 X                    P[X <= x]
 1                        0.1
 2                        0.3
 3                        0.7
 4                        1.0