Final answer:
The cumulative probability distribution of a random variable X, or cumulative distribution function (cdf), is a function representing the probability that X is less than or equal to a specific value x. It sums probabilities for discrete random variables or integrates the probability density function for continuous ones. It is used to calculate probabilities for specific intervals and the complements of events in probability distributions.
Step-by-step explanation:
The cumulative probability distribution of a random variable X, often represented as the cumulative distribution function (cdf), is defined as P(X ≤ x). It is a function that represents the probability that the random variable X will take on a value less than or equal to a specific value x. For discrete random variables, this involves summing up the probabilities of all outcomes less than or equal to x. In the case of continuous random variables, the cdf is found by integrating the probability density function (pdf) up to x, which gives us the area under the curve from the lowest possible value of X to x.
Importantly, the cumulative distribution function can also be used to calculate probabilities for intervals and complements. For instance, the probability P(c < X < d) can be calculated using the cdf to find the probability that X lies within a certain interval (c, d). To find the complement probability P(X > x), one would calculate 1 - P(X ≤ x) for continuous distributions, since the total probability must sum to 1.
A cumulative distribution function must satisfy two key properties for a discrete probability distribution: the probability for each outcome must be between 0 and 1, and the sum of all probabilities must equal 1. For example, if X takes on the values 0, 1, 2, 3, 4, 5 with respective probabilities that sum up to 1, it is a valid discrete probability distribution.