.. title: Engineering Probability Class 16 Mon 2018-03-19
.. slug: class16
.. date: 2018-03-18
.. tags: mathjax
.. category: class
.. link: 
.. description: 
.. type: text

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


Review of normal 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.

#. Iclicker:   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
   #. .58
   #. .84

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

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

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

#. See intro I did in last class.

#. 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.   

#. Example 5.5 on page 238.

#. Example 5.6 on page 240.

#. Example 5.7 on page 241.

#. Example 5.8 on page 242.


Next time
---------

#. 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]?

#. Review Extend section 5.3.1 example 5.14 on page 247.

#. Independence: Example 5.22 on page 256.  Are 2 normal r.v. independent for different values of $\\rho$ ?

#. Example 5.31 on page 264.  This is a noisy comm channel, now with Gaussian (normal) noise.  The problems are:

   #. what input signal to infer from each output, and
   #. how accurate is this?

#. 5.6.2 Joint moments etc

   #. Work out for 2 3-sided dice.
   #. Work out for tossing dart onto triangular board.

#. Example 5.27: correlation measures ''linear dependence''.  If the dependence is more complicated, the variables may be dependent but not correlated.

#. Covariance, correlation coefficient.

#. Section 5.7, page 261. Conditional pdf.  There is nothing majorly new here; it's an obvious extension of 1 variable.

   #. Discrete: Work out an example with a pair of 3-sided loaded dice.

   #. Continuous: a triangular dart board.  There is one little trick because for P[X=x]=0 since X is continuous, so how can we compute P[Y=y|X=x] = P[Y=y &amp; X=x]/P[x]?  The answer is that we take the limiting probability P[x<X<x+dx] etc as dx shrinks, which nets out to using f(x) etc.

#. Example 5.31 on page 264.  This is a noisy comm channel, now with Gaussian (normal) noise.  This is a more realistic version of the earlier example with uniform noise.  The application problems are:

   #. what input signal to infer from each output,
   #. how accurate is this, and
   #. what cutoff minimizes this?

   In the real world there are several ways you could reduce that error:

   #. Increase the transmitted signal,
   #. Reduce the noise,
   #. Retransmit several times and vote.
   #. Handshake: Include a checksum and ask for retransmission if it fails.
   #. Instead of just deciding X=+1 or X=-1 depending on Y, have a 3rd decision, i.e., *uncertain* if $|Y|<0.5$, and ask for retransmission in that case.  

#. Section 5.8 page 271: Functions of two random variables.

   #. We already saw how to compute the pdf of the sum and max of 2 r.v.

#. What's the point of transforming variables in engineering?  E.g. in video, (R,G,B) might be transformed to (Y,I,Q) with a 3x3 matrix multiply.  Y is brightness (mostly the green component).  I and Q are approximately the red and blue.  Since we see brightness more accurately than color hue, we want to transmit Y with greater precision.  So, we want to do probabilities on all this.

#. Functions of 2 random variables

   #. This is an important topic.
   #. Example 5.44, page 275. Tranform two independent Gaussian r.v from
      (X,Y) to (R, $\\theta$} ).  
   #. Linear transformation of two Gaussian r.v.  
   #. Sum and difference of 2 Gaussian r.v. are independent.

#. Section 5.9, page 278: pairs of jointly Gaussian r.v.

   #. I will simplify formula 5.61a by assuming that $\\mu=0, \\sigma=1$.

      $$f_{XY}(x,y)= \\frac{1}{2\\pi \\sqrt{1-\\rho^2}} e^{ \\frac{-\\left( x^2-2\\rho x y + y^2\\right)}{2(1-\\rho^2)} }  $$ .

   #. The r.v. are probably dependent.  $\\rho$} says how much.
   #. The formula degenerates if $|\\rho|=1$ since the numerator and denominator are both zero.  However the pdf is still valid.  You could make the formula valid with l'Hopital's rule.
   #. The lines of equal probability density are ellipses.
   #. The marginal pdf is a 1 variable Gaussian.

#. Example 5.47, page 282: Estimation of signal in noise

   #. This is our perennial example of signal and noise.  However, here the signal is not just $\\pm1$ but is normal.  Our job is to find the ''most likely'' input signal for a given output.

#. Next time: We've seen 1 r.v., we've seen 2 r.v.  Now we'll see several r.v.



Material added after class
--------------------------

#. `My handwritten tablet notes <../../handwritten/319.pdf>`_.