What is a cumulative distributive function?

Answer 1

It the the probability that the random variable in question takes any value up to and including the argument.

Suppose you have a random variable X and f(x) is the probability that X = x [that is, the rv X takes the value x].

If F(x) denotes the cumulative distribution function of X, then

F(x) is the sum of all f(y) where y <= x.

Thus, for a fair die,

F(1) = f(1) = 1/6

F(2) = f(1) + f(2) = 2/6

F(3) = f(1) + f(2) + f(3) = 3/6 and so on.

Note that

F(X) = 0 for X < 1,

F(a+b) where a is an integer in the interval [1,6] and 0<b<1 is F(a). Thus, for example, F(3.5) = F(3).

and F(x) = 1 for x >=6.

In the case of continuous probability distributions, the summation is replaced by integration.

Answer 2

If a random variable X has probability distribution function f(x) then the cumulative distribution function is

F(x) = p(X ≤ x).

For discrete function, this is the sum of the probabilities that X takes a value ≤ x while for a continuous function it is the integral of f(x) (with respect to x) from -∞ to x.