Cumulative distribution function

The basic stuff of probability theory consists of events and their probability measures. Given a random variable X, consider the event {X ≤ x}; incidentally, note how we use x to denote a number. The probability of this event is a function of x.

DEFINITION 6.3 (Cumulative distribution function) Let X be a random variable. The function

for , is called cumulative distribution function (CDF).

The notation F_X(x) clarifies that the CDF is a function of x, associated with the random variable X, as suggested by the attached subscript. In passing, we may also note that the definition above is quite general, and it does not require that X is a discrete random variable; the CDF can be defined like this for continuous random variables as well. To build intuition, let us see how the CDF compares to cumulative relative frequencies in descriptive statistics.

Example 6.3 Let us consider a discrete random variable naturally related to dice throwing and build its CDF. To begin with, it is not possible to get a number strictly smaller than one. Hence

Then, there is a jump in the function when we consider x = 1:

If we increase x, the CDF will stay there as long as 1 ≤ x < 2, since the discrete random variable cannot take any value in the open interval (1, 2). Then, there is another jump for x = 2:

Fig. 6.1 CDF for dice throwing.

since the event {X ≤ 2} includes the events {X = 1} and {X = 2}. Similar jumps occur for x = 3,4,5. Finally, we have

For x > 6, the function will just stay there, since the support is bounded by x = 6. The resulting CDF is depicted in Fig. 6.1. Note that the figure is drawn according to the standard convention of discontinuous functions.³ If we are a bit picky, we should say that the function is continuous from the right, but discontinuous from the left. In other words, , but if we approach x = 1 from the left, i.e, the value of the function we see is 0:

The plot in Fig. 6.1 suggests a few properties of the CDF for a discrete random variable:

1. The CDF is a nondecreasing function:To see this, observe that event E₁ = {X ≤ x₁} is a subset of event E₂ = {X ≤ x₂}, which implies P(E₁) ≤ P(E₂).
2. If we consider the support x_i, i =0 1, 2, 3,…, where x_i < x_i+1, we haveTable 6.1 A discrete probability distribution.Indeed, the value of the CDF is a probability, so it must stay within the interval [0, 1].
3. The CDF for a discrete random variable is a piecewise constant function, with jumps corresponding to values included in the support.