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\flushleft\underline{Lab 6 --- MATH 1586 Spring 2018} \\
Recall: this site plots surfaces nicely: \href{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/}{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/} \\
Also, it is helpful to use \href{http://www.wolframalpha.com/}{WolframAlpha} to compute integrals. You may also of course use Mathcad to do it. \\

This lab concerns the application of double integrals to probability theory. Let $D$ be a region in the plane (called the domain) and let $f(x,y)$ be a function which describes a surface above $D$. We say that $f$ is a probability density function if the following two properties hold:
\begin{enumerate}[i.)]
\item for every point $(x,y)$ in $D$, $f(x,y) \geq 0$, and
\item $\displaystyle\iint_D f(x,y) \mathrm{d}A=1$. 
\end{enumerate}
A set $E$ inside of $D$ is called an event. It is often asked in probability theory to find the ``probability of an event" occurring. This is what we will calculate in this lab.

\begin{enumerate}[1.]
\item Consider the ``normal" bivariate probability density function ``with mean $0$ and standard deviation $1$" given by $f(x,y)=\dfrac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$ with domain $D=\mathbb{R}^2$ (the whole plane). Consider the event defined by $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$. Attach a picture of the distribution over this region and also find the probability of this event.

\item Use the distribution in problem $1$ to find the probability that $x \leq y$ in the region $E$. To do this, draw the curve $y=x$ in the region with $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$. Shade the half of this drawing that agrees with ``$x \leq y$" in. Set up the integral over this region (it should be triangular; draw $y=x$ and then figure out which of the two halves obeys $x \leq y$) and then use that to find the probability.

\item Consider the domain $D$ determined by $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$ and the distribution $f(x,y)=\dfrac{3}{8} (x^2+y^2)$. Plot a picture of this distribution and find the probability that $y \leq x$ in the region.
\end{enumerate}

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