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\centerline{Mathematics 136 -- Calculus 2}
\centerline{More on Distribution Functions (pdfs)}
\centerline{March 17, 2020}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
A function $f(x)$ that is non-negative for all $x$ and that satisfies the condition
that \emph{the total area between the graph $y = f(x)$ and the $x$-axis is one}, i.e. 
\begin{equation}
\label{totalint}
\int_{-\infty}^\infty f(x)\ dx = 1
\end{equation}
is called a \emph{probability density function} (pdf).  A random variable $X$ is
said to have pdf $f(x)$ (or have a distribution described by $f(x)$) if the probability
that $a \le X \le b$ is given by 
\begin{equation}
\label{prob}
P(a \le X \le b) = \int_a^b f(x)\ dx
\end{equation}
for all $a,b$. 

The \emph{mean, or expected value} of a random variable $X$ with distribution described
by the pdf $f(x)$ is computed by 
\begin{equation}
\label{evalue}
E(X) = \int_{-\infty}^\infty x\cdot f(x)\ dx,
\end{equation}
provided that this improper integral exists.  This can be thought of as something
like the \emph{weighted average} of $x$, with weights given by the pdf $f(x)$.  
 
Note that the integrals in \eqref{totalint}, \eqref{evalue} are \emph{improper integrals}, 
and the integral in \eqref{prob} can be improper as well.  This means that we will need
to apply the techniques we learned for improper integrals to compute them if we need to.  
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\noindent
{\it Questions}
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\begin{enumerate}
\item[A)]  Find the constant $c$ so that the function 
$$f(x) = \begin{cases} c e^{-3x} & x \ge 0\\
                           0 & x < 0
\end{cases}$$ 
is a pdf.  Use your value of $c$ to determine $P(1 < X < 3)$ if $X$ is a 
random variable with a distribution described by this pdf.  (These are
called \emph{exponential} pdfs.  They are used to model the distributions
of random quantities like
the lifetime of a randomly selected electronic component of a particular type, 
or the time between random incoming calls to customer service call centers, etc.)
\item[B)]  Find the expected value of $X$ from question A.  What is the probability
that $X > E(X)$?  What is the value $x_m$ where $P(X > x_m) = \frac{1}{2}$ (the median)?
How are the mean and median related in this example?
\item[C)]  Find the $c$ so that $f(x) = \displaystyle{\frac{c}{x^2 + 1}}$ (domain
all real numbers) is a pdf.
\item[D)]  Does the expected value exist for a random variable with pdf as in part C?
Why or why not?  Does the median exist?  
\end{enumerate}
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