Definitions for Continuous Random Variables

Tags
Statistics
Date
Oct 9, 2019
Description
The probability density function, cumulative distribution function, expectation and variance for a continuous random variable.
Slug
mathematical-statistics-continuous-rv-definition
In
Definitions for Discrete Random Variables
, we mainly focused on discrete random variables whose set of possible values is either finite or countably infinite. In this chapter, we study random variables whose set of possible values is uncountable. We’ll see later on that a lot of the cases we’ve discussed have analogs in the continuous case.

Probability density function

The continuous random variable is a random variable with infinite possible outcomes (a subset of the real line). We say is a continuous random variable if there exists a nonnegative function defined for all , having the property that for any set of real numbers,
Here the function is called the probability density function (PDF). It resembles the probability mass function in the discrete case. The PDF has the following properties:
1. .
1. .
1. .
We can also define the cumulative distribution function (CDF) for a continuous random variable:

Properties example

These properties often come in handy when we have unknown quantities in a PDF. Suppose is a continuous random variable with probability density function
and we’d like to find as well as the probability .
Now we can easily find . With the PDF given, it’s trivial to find the CDF:

Suppose , the lifetime of an item, is a continuous random variable with a density function
What is the probability that the item functions between 50 and 150 days?
In mathematical terms, we want to calculate . We first need to find the value of .
Recall that , and because the derivative is a linear function.
With , we can calculate

Expectation and variance

Earlier we’ve defined the expectation for discrete random variables. If is a continuous random variable with probability density function , we have
so it’s easy to find the analog for the expectation of to be
Similarly, the expected value of a real-valued function of is
which can be used to derive the variance of :
Suppose is a continuous random variable with density function