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.)

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.)

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

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.

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