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Various exercises relating to probability.

• If it exists, find the value of the constant $c$ below which makes the function a PDF on the interval $[e,e^2]$. 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 $c$ which makes it a PDF on the interval $[e^{-2},e^2]$. If no such constant exists, enter N/A.
The function $\ln x$ is negative on $[e^{-2},1)$ and positive on $(1,e^2]$. What does that imply for the possibility of it being a PDF?
Suppose $X$ 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 $c$. If the mean hold time is $2$ 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 $c$.
The equation will be
This gives $c = 1/2$.
The easiest formula to use here is where $\mu$ is the mean.
Suppose $X$ is a random variable on the interval $[1,10]$ whose PDF is given by for some positive constant $c$ (note: this is the PDF that arises in the phenomenon known as Benford’s Law). Compute the mean $\mu$ and the median $m$ 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 $x$to the right/for larger $x$ , it is expected (and in fact, true) that the mean is largersmaller than the median.
First you need to solve for the constant $c$.

### Sample Quiz Questions

Find the value of $c$ which makes the function a probability density function on the interval $[0, \infty )$. What is the value of the mean $\mu$ of the corresponding random variable? (Hints will not be revealed until after you choose a response.)

$\displaystyle c = 1, ~ \mu = \frac {1}{2}$ $\displaystyle c = -1, ~ \mu = \frac {3}{4}$ $\displaystyle c = 1, ~ \mu = 1$ $\displaystyle c = -1, ~ \mu = \frac {5}{4}$ $\displaystyle c = 1, ~ \mu = \frac {3}{2}$ $\displaystyle c = -1, ~ \mu = \frac {7}{4}$
A certain random variable $X$ takes values in the interval $\left [2 \pi , \frac {5}{2} \pi \right ]$. If the probability density function is given by for some appropriate value of the constant $A$, compute the expected value $\mu$ of $X$. (Hints will not be revealed until after you choose a response.)
$\displaystyle \mu = -1 + 2 \pi$ $\displaystyle \mu = 1 + \frac {3}{2} \pi$ $\displaystyle \mu = 2 \pi$ $\displaystyle \mu = -1 + \frac {5}{2} \pi$ $\displaystyle \mu = 1 + 2 \pi$ $\displaystyle \mu = \frac {5}{2} \pi$

### Sample Exam Questions

A certain random variable $X$ has values in $(1,\infty )$ and has the property that there is some constant $C$ such that for every $a > 1$. Compute the value of $C$ and determine whether the expected value $\mu$ of $X$ is finite or infinite. [Hint: There is enough information given to compute $C$ without calculating any integrals.]
$C = \ln 2$ and $\mu < \infty$ $C = 1$ and $\mu < \infty$ $C = (\ln 2)^{-1}$ and $\mu < \infty$ $C = \ln 2$ and $\mu = \infty$ $C = 1$ and $\mu = \infty$ $C = (\ln 2)^{-1}$ and $\mu = \infty$

The function is a probability density function for a certain value of $k$. For that probability density function, find the probability that $x > 2$.

$\displaystyle \frac {1}{2}$ $\displaystyle \frac {1}{3}$ $\displaystyle \frac {1}{4}$ $\displaystyle \frac {2}{3}$ $\displaystyle \frac {1}{5}$ $\displaystyle \frac {1}{6}$

For a certain real number $k$, the function is a probability density function for a continuous random variable $X$. For this value of $k$, find the probability that $X > 1$.

$0$ $\displaystyle \frac {1}{3}$ $\displaystyle \frac {2}{3}$ $1$ $\displaystyle \frac {1}{2}$ $\displaystyle \frac {1}{4}$

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

$\displaystyle C = \frac {b^3}{4}$, mean $\displaystyle = b$ $\displaystyle C = \frac {4}{b^2}$, mean $\displaystyle = b$ $\displaystyle C = \frac {4}{b}$, mean $\displaystyle = b^2$ $\displaystyle C = \frac {4}{b^3}$, mean $\displaystyle = \frac {3}{2}b$ $\displaystyle C = \frac {4}{b^2}$, mean $\displaystyle = \frac {3}{2}b^2$ $\displaystyle C = \frac {4}{b}$, mean $\displaystyle = \frac {3}{2}b^3$