.. title: Engineering Probability Class 15 Thu 2022-03-03
.. slug: class15
.. date: 2022-02-28
.. tags: class, exam
.. link: 
.. description: 
.. type: text
.. has_math: true

.. sectnum::
.. contents:: Table of contents::
..



Matlab
------

#. Matlab, Mathematica, and Maple all will help you do
   problems too big to do by hand.  Today I'll demo Matlab.

#. Matlab

   #. Major functions::

	cdf(dist,X,A,...)
	pdf(dist,X,A,...) 

   #. Common cases of dist (there are many others)::

	'Binomial'
	'Exponential'
	'Poisson'
	'Normal'
	'Geometric'
	'Uniform'
	'Discrete Uniform' 

   #. Examples::

	pdf('Normal',-2:2,0,1)
	cdf('Normal',-2:2,0,1)

	p=0.2
	n=10
	k=0:10
	bp=pdf('Binomial',k,n,p)
	bar(k,bp)
	grid on

	bc=cdf('Binomial',k,n,p)
	bar(k,bc)
	grid on

	x=-3:.2:3
	np=pdf('Normal',x,0,1)
	plot(x,np)

   #. Interactive GUI to explore distributions:  disttool
   #. Random numbers::

	rand(3)
	rand(1,5)
	randn(1,10)
	randn(1,10)*100+500
	randi(100,4)

   #. Interactive GUI to explore random numbers:  randtool
   #. Plotting two things at once::

	x=-3:.2:3
	n1=pdf('Normal',x,0,1)
	n2=pdf('Normal',x,0,2)
	plot(x,n1,n2)
	plot(x,n1,x,n2)
	plot(x,n1,'--r',x,n2,'.g')

#. Use Matlab to compute a geometric pdf w/o using the builtin function.

#. Review.  Which of the following do you prefer to use?

   a. Matlab
   #. Maple
   #. Mathematica
   #. Paper.  It was good enough for Bernoulli and Gauss; it's good enough for me.
   #. Something else (please email about it me after the class).


My opinion
==========

This is my opinion of Matlab.

#. Advantages

   #. Excellent quality numerical routines.
   #. Free at RPI.
   #. Many toolkits available.
   #. Uses parallel computers and GPUs.
   #. Interactive - you type commands and immediately see results.
   #. No need to compile programs.

#. Disadvantages

   #. Very expensive outside RPI.
   #. Once you start using Matlab, you can't easily move away when their prices rise.
   #. You must force your data structures to look like arrays.
   #. Long programs must still be developed offline.
   #. Hard to write in Matlab's style.
   #. Programs are hard to read.

#. Alternatives

   #. Free clones like Octave are not very good.
   #. The excellent math routines in Matlab are also available free in C++ librarues
   #. With C++ libraries using template metaprogramming, your code looks like Matlab.
   #. They compile slowly.
   #. Error messages are inscrutable.
   #. Executables run very quickly.



Tutorial on probability density
-------------------------------

Since the meaning of probability density when you transform variables is
still causing problems for some people, think of changing units from
English to metric.  First, with one variable, X.

#. Let X be in feet and be U[0,1]. 

   $$f_X(x) =  \\begin{cases} 1& \\text{if }  0\\le x\\le1\\\\ 0&\\text{otherwise} \\end{cases}$$
#. $P[.5\\le x\\le .51] = 0.01$.
#. Now change to centimeters.  The transformation is $Y=30X$.
#. $$f_Y(y) =  \\begin{cases} 1/30 & \\text{if }  0\\le y\\le30\\\\ 0&\\text{otherwise} \\end{cases}$$
#. Why is 1/30 reasonable?
#. First, the pdf has to integrate to 1:  $$\\int_{-\\infty}^\\infty f_Y(y) =1$$
#. Second, $$\\begin{align} & P[.5\\le x\\le .51]  \\\\ &= \\int_.5^.51 f_X(x) dx  \\\\& =0.01  \\\\& =  P[15\\le y\\le 15.3]  \\\\& = \\int_{15}^{15.3} f_Y(y) dy  \\end{align}$$


Tutorial on probability density - 2 variables
---------------------------------------------

Here's a try at motivating changing 2 variables.

#. We're throwing darts uniformly at a one foot square dartboard.

#. We observe 2 random variables, X, Y, where the dart hits (in Cartesian coordinates).

#. $$f_{X,Y}(x,y) =  \\begin{cases} 1& \\text{if}\\,\\,  0\\le x\\le1 \\cap 0\\le y\\le1\\\\ 0&\\text{otherwise} \\end{cases}$$

#. $$P[.5\\le x\\le .6 \\cap .8\\le y\\le.9]  = \\int_{.5}^{.6}\\int_{.8}^{.9} f_{XY}(x,y) dx \\, dy = 0.01 $$

#. Transform to centimeters:  $$\\begin{bmatrix}V\\\\W\\end{bmatrix} = \\begin{pmatrix}30&0\\\\0&30\\end{pmatrix} \\begin{bmatrix}X\\\\Y\\end{bmatrix}$$

#. $$f_{V,W}(v,w)   = \\begin{cases} 1/900& \\text{if } 0\\le v\\le30 \\cap 0\\le w\\le30\\\\ 0&\\text{otherwise} \\end{cases}$$

#. $$P[15\\le v\\le 18 \\cap 24\\le w\\le27]  = \\\\ \\int_{15}^{18}\\int_{24}^{27} f_{VW}(v,w)\\, dv\\, dw = \\frac{ (18-15)(27-24) }{900} =  0.01$$

#. See Section 5.8.3 on page 286.




Comic
----------

`Dilbert <../files/comics/dilbert-random.png>`_