Cumulative Probability refers to the probability that a certain event occurs less than or equal to a certain number of times.

It is obtained by adding up the probabilities of the random variable taking values less than or equal to the specified number.

**Example:**

Suppose that two coins are tossed. Here the possible events are {HH, HT, TH, TT}. Let X denote the number of heads obtained.

The probability of X=1, which is getting one head is 2/4 = 0.5 since we have two favorable outcomes (HT, TH) out of four. So we obtain the probability distribution table as follows,

X = number of Heads | Probability P(X=x) |

0 | 1/4 = 0.25 |

1 | 2/4 = 0.5 |

2 | 1/4 = 0.25 |

We can obtain the cumulative probability for a particular value of X by adding up the probabilities of the current and previous values of X.

For example, P(X≤1) = P(X=0) + P(X=1) = 0.25 + 0.5 = 0.75.

In a similar manner, we obtain the cumulative probability distribution as follows:

X = number of Heads | Probability P(X=x) | Cumulative Probability P(X≤x) |

0 | 1/4 = 0.25 | 0.25 |

1 | 2/4 = 0.5 | 0.25+0.5 = 0.75 |

2 | 1/4 = 0.25 | 0.25+0.5+0.25 = 1 |

### How do you calculate the cumulative probabilities for a continuous distribution?

As seen above we can calculate the cumulative probabilities for a discrete random variable by adding up all the previous probabilities.

Cumulative Probability = * Σ*_{X≤x} P(X=x). Here the sum is taken over all values of X less than x.

For a continuous random variable, given the probability distribution function (pdf) f(x) we can obtain the cumulative probability distribution function F(x) by integrating the pdf from minus infinity to X.

Cumulative Probability = P(X≤x) = _{0}∫^{x} f(x)dx.

** Example**: Suppose that a continuous random variable X has the pdf,

f(x) = 3x^{2}, where 0≤ x ≤ 1.

Find the cumulative probability distribution function.

** Solution**:

F(x) = P(X≤ x) = _{0}∫^{x} f(x)dx = = x^{3} which is the required cumulative probability distribution function.