Let be a probability space and a random variable. Define a function as the probability takes on a value less than or equal to .

That is, is the measure of the set of such that . Notice that is an increasing function as implies and so

And since a measure is monotonic (i.e., ), we have that .

Recall that if is a decreasing sequence of measurable sets (i.e., ), then , provided there exsists an with finite (in our case is a probability measure, so this is given to us anyway). We can use this fact to show that our function is right-continuous.

Let be a real-valued sequence that decreases to . Then we have that

This shows that and so is right-continuous.

Additionally, and .

The function that we’ve described is known as the cumulative distribution function (or just distribution function) of our random variable . If only takes on countably many values, then we say that is a discrete random variable. If is a continuous function, we say is a continuous random variable.

Now that we’ve seen how to construct this distribution function given a random variable , the natural question to ask is if we have a function that satisfies the properties of a distribution that we’ve listed, is there a corresponding random variable?

Suppose that is an increasing, right-continuous function with and . Using the ideas from the Lebesgue-Stieltjes measure article, we have that gives us a measure and sigma-algebra on . Let and be the measure and sigma-algebra, respectively, that we get from . It follows from the fact that and that , and so is a probability space.

Define as . Note that this is a random variable as

To see that is a measurable function (random variable), let and . There exists a sequence of intervals that covers and

Define and ; note that and cover and , respectively. Also notice that . This gives us

so is a measurable function (random variable). This means that any function which is increasing, right-continuous, and and , is the distribution of some random variable.

### Like this:

Like Loading...

*Related*

## Leave a Reply