In this last application of integrals that we’ll be looking
at we’re going to look at probability.
Before actually getting into the applications we need to get a couple of
definitions out of the way.
Suppose that we wanted to look at the age of a person, the
height of a person, the amount of time spent waiting in line, or maybe the
lifetime of a battery. Each of these
quantities have values that will range over an interval of integers. Because of this these are called continuous random variables. Continuous random variables are often
represented by X.
Every continuous random variable, X, has a probability density
function, . Probability density functions satisfy the
following conditions.
 for all x.

Probability density functions can be used to determine the
probability that a continuous random variable lies between two values, say a and b. This probability is
denoted by and is given by,
Let’s take a look at an example of this.
Example 1 Let
for and for all other values of x. Answer each of the
following questions about this function.
(a) Show
that is a probability density function. [Solution]
(b) Find
[Solution]
(c) Find
[Solution]
Solution
(a) Show that is
a probability density function.
First note that in the range is clearly positive and outside of this
range we’ve defined it to be zero.
So, to show this is a probability density function we’ll
need to show that .
Note the change in limits on the integral. The function is only nonzero in these
ranges and so the integral can be reduced down to only the interval where the
function is not zero.
[Return to Problems]
(b) Find
In this case we need to evaluate the following integral.
So the probability of X
being between 1 and 4 is 8.658%.
[Return to Problems]
(c) Find
Note that in this case is equivalent to since 10 is the largest value that X can be. So the probability that X is greater than or equal to 6 is,
This probability is then 66.304%.
[Return to Problems]

Probability density functions can also be used to determine
the mean of a continuous random variable.
The mean is given by,
Let’s work one more example.
Example 2 It
has been determined that the probability density function for the wait in
line at a counter is given by,
where t is the
number of minutes spent waiting in line.
Answer each of the following questions about this probability density
function.
(a) Verify
that this is in fact a probability density function. [Solution]
(b) Determine
the probability that a person will wait in line for at least 6 minutes.\
[Solution]
(c) Determine
the mean wait in line. [Solution]
Solution
(a) Verify that this is in
fact a probability density function.
This function is clearly positive or zero and so there’s
not much to do here other than compute the integral.
So it is a probability density function.
[Return to Problems]
(b) Determine the
probability that a person will wait in line for at least 6 minutes.
The probability that we’re looking for here is .
So the probability that a person will wait in line for
more than 6 minutes is 54.8811%.
[Return to Problems]
(c) Determine the mean
wait in line.
Here’s the mean wait time.
So, it looks like the average wait time is 10 minutes.
[Return to Problems]
