.. title: Engineering Probability Class 16 Mon 2022-03-14
.. slug: class16
.. date: 2022-03-14
.. tags: class, exam
.. link: 
.. description: 
.. type: text
.. has_math: true

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



Review of normal (Gaussian) distribution
----------------------------------------

#. Review of the normal distribution.  If $\\mu=0,  \\sigma=1$ (to keep it simple), then:  $$f_N(x) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}} $$

#. Show that $\\int_{-\\infty}^{\\infty} f(x) dx =1$.  This is example 4.21 on page 168.

#. Review:   Consider a normal r.v. with    $\\mu=500,  \\sigma=100$.   What is the probability of being in the interval [400,600]?  Page 169 might be useful.

   a. .02
   #. .16
   #. .48
   #. .68
   #. .84

#. Repeat that question for the interval [500,700].

#. Repeat that question for the interval [0,300].   


Varieties of Gaussian functions
-------------------------------

#. Book page 167:  $\\Phi(x)$ is the CDF of the Gaussian.

#. Book page 168 and table on page 169: $Q(x) = 1 - \\Phi(x)$.

#. Mathematica (and other SW packages): *Erf[x]* is integral of pdf from 0 to x $Erf(x) = Q(x)-.5$ .

#. *Erfx(x) = 1-Erf(x)*.   

(The nice thing about standards is that there are so many of them.)


Chapter 5, Two Random Variables, 2
----------------------------------

#. Today's reading: Chapter 5, page 233-242.

#. Review: An *outcome* is a result of a random experiment.  It need not be a number.  They are selected from the *sample space*.   A *random variable* is a function mapping an outcome to a real number.  An *event* is an interesting set of outcomes.
   
#. Example 5.3 on page 235.   There's no calculation here, but this topic is used for several future problems.

#. Example 5.5 on page 238.  

#. Example 5.6 on page 240.  Easy, look at it yourself.

#. Example 5.7 on page 241. Easy, look at it yourself.

#. Example 5.8 on page 242. Easy, look at it yourself.

#. Example 5.9 on page 242.

#. 5.3 Joint CDF page 242.
   
#. Example 5.11 on page 245.  What is f(x,y)?

#. Example 5.12 p 246

#. Cdf of mixed continuous - discrete random variables: section 5.3.1 on page 247.  The input signal X is 1 or -1.  It is perturbed by noise N that is U[-2,2] to give the output Y..  What is P[X=1|Y<=0]?

#. Example 5.14 on page 247.

#. Example 5.16 on page 252.