Analysts are often interested in finding the probability of a range of outcomes rather than a specific outcome. A

The two characteristics are:

- The cumulative distribution function lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1.
- As we increase x, the cdf either increases or remains constant.

Given the cumulative distribution function, the probabilities for the random variable can also be calculated. In general:

A

Consider a probability function: p(X) = X/6 for X = 1, 2, 3 and p(X) = 0 otherwise. In a previous example it was shown that p(1) = 1/6, p(2) = 2/6, and p(3) = 3/6.

- F(1) indicates the probability that has been accumulated up to and including the point X = 1. Clearly, 1/6 of probability has been accumulated up to this point, so F(1) = 1/6.
- F(2) indicates the probability that has been accumulated up to and including the point X = 2. When X = 2 is reached, the accumulation of 1/6 is taken from X = 1 and 2/6 from X = 2; in total accumulation is 1/6 + 2/6 = 3/6 or, of the probability, so F(2) = 3/6.
- F(3) indicates the probability that has been accumulated up to and including the point X = 3. By the time X = 3 is reached, all the probability has been accumulated: 1/6 from X = 1, 2/6 from X = 2 and 3/6 from X = 3. Thus, 1/6 + 2/6 + 3/6 = 1. Therefore, F(3) = 1.

It is also possible to calculate F(X) for intermediate values. F(0) = 0, as no probability has been accumulated up to the point X = 0; F(1.5) = 1/6, as by the time X = 1.5 is reached, 1/6 of probability has been accumulated from X = 1; F(7) = 1, as by the time 7 is reached, all possible probability from X = 1, 2 and 3 has been collected.

sahilb7: F(X) is the cumulative sum of probabilities p(X) for all values less than X. |