.. title: Engineering Probability Class 13 Mon 2019-02-25
.. slug: class13
.. date: 2019-02-25
.. tags: mathjax
.. category: class
.. link: 
.. description: 
.. type: text

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

This year's exam 1 online
-------------------------

with and w/o answers.  See  `here <../../categories/>`_.

We gave full points even if you didn't finish the arithmetic to compute a number.   In the real world, you have computers.    However, in the real world, accurate analysis and computation matter.   In 1954, physicists made an eensy teensy error designing `Castle Bravo <https://en.wikipedia.org/wiki/Castle_Bravo>`_.


Homework 5
----------

online, due after break.

Notation
--------

How to parse $F_X(x)$

#. Uppercase F means that this is a cdf.   Different letters may indicate different distributions. 

#. The subscript X is the name of the random variable.

#. The x is an argument, i.e., an input.

#. $F_X(x)$ returns the probability that the random variable is less or equal to the value x, i.e. prob(X<=x).
   
   

Matlab
------

#. Matlab, Mathematica, and Maple all will help you do
   problems too big to do by hand.  Sometime I'll demo Matlab since IMO more of the
   class knows it.

#. 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.

#. Iclicker.  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.

      
Chapter 4 ctd
-------------

#. Taxi example: Sometimes there are mixed discrete and continuous r.v.   

   a. Let X be the time X to get a taxi at the airport.
   #. 80% of the time a taxi is already there, so p(X=0)=.8.
   #. Otherwise we wait a uniform time from 0 to 20 minutes, so p(a<x<b)=.01(b-a), for 0<a<b<20.


#. Iclicker.   For the taxi example, what is F(0)?

   a. 0
   #. .2
   #. .8
   #. .81
   #. 1

#. iclicker.   For the taxi example, what is F(1)?

   a. 0
   #. .8
   #. .81
   #. .9
   #. 1

#. Text 4.2 p 148 pdf

#. Simple continuous r.v. examples: uniform, exponential.

#. The **exponential** distribution complements the Poisson distribution.  The
   Poisson describes the number of arrivals per unit time.  The exponential
   describes the distribution of the times between consecutive arrivals.

   Ex 4.7 p 150: exponential r.v.

#. Properties

   a. Memoryless.

   #. $f(x) = \\lambda e^{-\\lambda x}$ if $x\\ge0$, 0 otherwise.

   #. Example: time for a radioactive atom to decay.

#. Ski p    4.2.1 for now.
   
#. The most common continuous distribution is the **normal** distribution.

#. 4.2.2 p 152. Conditional probabilities work the same with
   continuous distributions as with discrete distributions.

#. p 154.  Gaussian r.v.

   a. $$f(x) = \\frac{1}{\\sqrt{2\\pi} \\cdot \\sigma} e^{\\frac{-(x-\\mu)^2}{2\\sigma^2}}$$

   #. cdf often called $\\Psi(x)$

   #. cdf complement:

      a. $$Q(x)=1-\\Psi(x) = \\int_x^\\infty \\frac{1}{\\sqrt{2\\pi} \\cdot \\sigma} e^{\\frac{-(t-\\mu)^2}{2\\sigma^2}} dt$$

      #. E.g., if $\\mu=500, \\sigma=100$,

         i. P[x>400]=0.66
         #. P[x>500]=0.5
         #. P[x>600]=0.16
	 #. P[x>700]=0.02
	 #. P[x>800]=0.001

#. Text 4.3 p 156 Expected value	    

#. Skip the other distributions (for now?).