# Links on Matlab

An introduction to the MATLAB functions that will be used in this course.
Matlab primer

Getting started with Matlab

Matlab tutorial

A practical introduction to Matlab

## How to perform symbolic integration in Matlab?

**Note:
Matlab is designed for numerical computation. What I shall describe below is a relatively
advanced feature in Matlab. If you don't know Matlab, visit the links above to familiarize yourself
with the basic of Matlab first. Otherwise, you may be confused.**

You need to make sure you have the symbolic toolbox before you can try the following. This is available in the Matlab installation in MSU.

### Using symbolic toolbox

Suppose we are given the data density

p(x) \propto exp( -(x-u)^2 / s^2 ) for x in (-\infty, \infty).

What is the analytic expression for this density?

You probably have noticed that p(x) is a Gaussian pdf, and the proportionality
constant can be looked up in the textbook. Let's try to compute this in Matlab instead.

First, we need some declarations.

>> syms x u real
>> syms s positive

This declares both x and u as **symbolic** real numbers, and s as a **symbolic** positive real number. Note that
normal variables in Matlab do not require declaration. Only symbolic ones do.

Next, we created the symbolic variable "f" to represent to the above density.

>> f = exp( -(x-u)^2 / s^2 )
f =
exp((x-u)^2/s^2)

We now need to integrate the expression with respect to x. Since
x ranges from negative infinity to positive infinity, the integration limit should be
-inf, +inf.

g = int(f, x, -inf, inf)
g =
s*pi^(1/2)

So, the normalization for the pdf is 1/(s*pi^(1/2)). In other words, the pdf for x should be

p(x) = 1/(s*pi^(1/2)) * exp( -(x-u)^2 / s^2 ) for x in (-\infty, \infty).

We can check this is indeed a valid pdf because it integrates to one:

>> pdf = 1/(s*sym(pi)^(1/2)) * exp( -(x-u)^2 / s^2 ) ;
>> int(pdf, x, -inf, inf)
ans =
1