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Mathematical Expression Editor

Various exercises relating to probability.

If it exists, find the value of the constant below which makes the function a
PDF on the interval . If no such constant exists, enter N/A.

You’ll need to
integrate by parts.

For the same function as above, find the value of the constant which makes it a
PDF on the interval . If no such constant exists, enter N/A.

The function is
negative on and positive on . What does that imply for the possibility of it
being a PDF?

Suppose is a random variable which represents the length of time your customers
remain on hold before reaching a customer service agent. A reasonable model for such
a random variable is to use an exponential distribution, i.e., to have a PDF given by
for some positive constant . If the mean hold time is minutes, what is the standard
deviation? What is the probability that a customer will have to wait more than one
standard deviation beyond the mean before their call is answered?

First you need to
solve for .

The equation will be

This gives .

The easiest formula to use here is
where is the mean.

Suppose is a random variable on the interval whose PDF is given by for some
positive constant (note: this is the PDF that arises in the phenomenon known as
Benford’s Law). Compute the mean and the median associated to this PDF. Which
is larger? The meanmedian is more sensitive to tails. Since this distribution has a tail extending to the left/for
smaller to the right/for larger , it is expected (and in fact, true) that the mean is largersmaller than the median.

First you need to solve for the constant .

Sample Quiz Questions

Find the value of which makes the function a probability density function on the
interval . What is the value of the mean of the corresponding random variable?
(Hints will not be revealed until after you choose a response.)

To compute the constant , we use the fact that the integral of a probability density
function must equal , so This gives the equation which then implies that .

To compute the mean , we use the formula Calculating the integral gives

A certain random variable takes values in the interval . If the probability density
function is given by for some appropriate value of the constant , compute the
expected value of . (Hints will not be revealed until after you choose a response.)

The constant will be the reciprocal of the integral

One can check that

To compute
the expected value we also need to compute the integral To compute the integral,
we can use integration by parts. A reasonable strategy is to integrate and
differentiate .

This gives the equality Therefore Therefore the expected value is the
ratio of the integrals, i.e.,

Sample Exam Questions

A certain random variable has values in and has the property that there is some
constant such that for every . Compute the value of and determine whether the
expected value of is finite or infinite. [Hint: There is enough information given to
compute without calculating any integrals.]

and and and and and and

We know that is always greater than one, so which gives . If we let denote the
probability density function of , then Differentiating both sides with respect to gives
so The expected value of must equal This integral will be finite by direct
comparison to the convergent integral .

The function is a probability density function for a certain value of . For that
probability density function, find the probability that .

For a certain real number , the function is a probability density function for a
continuous random variable . For this value of , find the probability that .

Let Find so that this is a probability density function (pdf) for the random variable
. Here is a positive constant. This function is used to model the distance between the
nucleus and the electron in a hydrogen atom. The constant is called the Bohr length.
Find the mean of the pdf.