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\layout Title

Integrals are not Scarey
\layout Author

Paul E.
 Johnson
\layout Section

The Integral 
\layout Standard

Even if you never took a calculus class, you may still have to read a book
 that uses integrals.
 The concept is not complicated, and as long as you don't actually have
 to solve some integrals, I expect a conceptual understanding is good enough
 for most people.
\layout Standard

The elongated S symbol represents integration.
 
\begin_inset Formula \[
\int_{a}^{b}f(x)dx\]

\end_inset 


\layout Standard

This means 
\begin_inset Quotes eld
\end_inset 

the total area under the curve 
\begin_inset Formula $f(x)$
\end_inset 

 between 
\begin_inset Formula $a$
\end_inset 

 and 
\begin_inset Formula $b$
\end_inset 

.
\begin_inset Quotes erd
\end_inset 

 Consider this.
\layout Standard


\begin_inset Graphics
	filename integral1.eps
	width 4in
	keepAspectRatio

\end_inset 


\layout Standard

The symbol 
\begin_inset Formula $dx$
\end_inset 

 represents the 
\begin_inset Quotes eld
\end_inset 

dummy variable of integration.
\begin_inset Quotes erd
\end_inset 

 It is a signal that you are supposed to move along the 
\begin_inset Formula $x$
\end_inset 

 axis when you sum up from 
\begin_inset Formula $a$
\end_inset 

 to 
\begin_inset Formula $b$
\end_inset 

.
\layout Standard

Sometimes people will describe the integral as a sum of really small slices
 out of 
\begin_inset Formula $f(x)$
\end_inset 

.
\layout Section

Integrals & Probability
\layout Standard

If you are studying a continuous random variable, 
\begin_inset Formula $x$
\end_inset 

, it means you are studying a variable that can take on real values in some
 domain, 
\begin_inset Formula $X$
\end_inset 

.
 Suppose the 
\begin_inset Quotes eld
\end_inset 

endpoints
\begin_inset Quotes erd
\end_inset 

 of 
\begin_inset Formula $X$
\end_inset 

 are 
\begin_inset Formula $left$
\end_inset 

 and 
\begin_inset Formula $right$
\end_inset 

, where 
\begin_inset Formula $left$
\end_inset 

 and 
\begin_inset Formula $right$
\end_inset 

 can take on any real value, as well as infinity.
 
\layout Standard

Real number digression: The symbol for the 
\begin_inset Quotes eld
\end_inset 

real number line
\begin_inset Quotes erd
\end_inset 

 is 
\begin_inset Formula $\mathbb{R}$
\end_inset 

.
 The set of all real numbers can be formally defined, but most of the time
 I just think of it as 
\begin_inset Quotes eld
\end_inset 

all numbers that can be written down as numbers with decimals, including
 infinite digits after the decimal point.
\begin_inset Quotes erd
\end_inset 

 
\layout Standard

If 
\begin_inset Formula $f(x)$
\end_inset 

 is a 
\series bold 
probability density function
\series default 
 (pdf) representing a probability distribution, it means these 3 conditions
 are true:
\layout Enumerate


\begin_inset Formula $f(x)\geq0$
\end_inset 

 for all 
\begin_inset Formula $x\in X$
\end_inset 


\layout Enumerate


\begin_inset Formula $f(x)\leq1$
\end_inset 

 for all 
\begin_inset Formula $x\in X$
\end_inset 


\layout Enumerate


\begin_inset Formula $\int_{left}^{right}f(x)dx=1$
\end_inset 

.
\layout Standard

There is always some 
\begin_inset Quotes eld
\end_inset 

tricky business
\begin_inset Quotes erd
\end_inset 

 about the probability of a particular, individual point.
 The probability that a particular point will occur is 0 because a point
 is a thing with no 
\begin_inset Quotes eld
\end_inset 

width.
\begin_inset Quotes erd
\end_inset 

 The probability that you could observe a particular outcome 
\begin_inset Formula $c$
\end_inset 

 is
\begin_inset Formula \[
\int_{c}^{c}f(x)dx\]

\end_inset 

 And that is always 0 by definition.
 So we are restricted to talking about outcomes in a particular range, say
 between 
\begin_inset Formula $c$
\end_inset 

 and 
\begin_inset Formula $d$
\end_inset 

.
 
\begin_inset Formula \[
\int_{c}^{d}f(x)dx\]

\end_inset 


\layout Standard

The 
\series bold 
cumulative distribution function
\series default 
 (cdf) is the probability that the outcome of a random draw will be smaller
 than some given value, say 
\begin_inset Formula $k$
\end_inset 

.
 It is often symbolized by a capital letter corresponding to the pdf, in
 this case 
\begin_inset Formula $F(k)$
\end_inset 

.
 Formally, it is the integral from 
\begin_inset Formula $left$
\end_inset 

 up to the point 
\begin_inset Formula $k$
\end_inset 

.
\begin_inset Formula \[
F(k)=\int_{left}^{k}f(x)dx\]

\end_inset 

 
\layout Section

Integrals for Multivariate Probability
\layout Standard

If the domain of outcomes is 2 or more dimensional, then the integral extends
 to represent it.
 For example, with 2 dimensions, 
\begin_inset Formula $X=\{ left_{x},right_{x}\}$
\end_inset 

 and 
\begin_inset Formula $Y=\{ left_{y},right_{y}\}$
\end_inset 

.
 The pdf is 
\begin_inset Formula $f(x,y)$
\end_inset 

 and the probability that an observation will occur in a region in which
 
\begin_inset Formula $x\in\{ c_{x},d_{x}\}$
\end_inset 

 and 
\begin_inset Formula $y\in\{ c_{y},d_{y}\}$
\end_inset 

 is represented as 
\begin_inset Formula \[
\int_{c_{x}}^{d_{x}}\int_{c_{y}}^{d_{y}}f(x,y)dxdy\]

\end_inset 


\layout Standard

There's a theorem that say the order of integration does not affect the
 value, so you can swap the 
\begin_inset Formula $x$
\end_inset 

 and 
\begin_inset Formula $y$
\end_inset 

 things in there.
\layout Standard

Generally, it is difficult to solve multivariate probability distributions,
 so we go searching about for simplifying results, such as independence,
 
\begin_inset Formula \[
f(x,y)=g(x)\cdot h(y)\]

\end_inset 


\layout Standard

This means that the joint observation of the pair 
\begin_inset Formula $(x,y)$
\end_inset 

 is just as likely as the separate observations of 
\begin_inset Formula $x$
\end_inset 

 and 
\begin_inset Formula $y$
\end_inset 

 .
\layout Standard

[stopped here, will work on more next season]
\the_end