.. title: Engineering Probability Class 18 Thurs 2021-04-01
.. slug: class18
.. date: 2021-04-01
.. tags: class
.. link: 
.. description: 
.. type: text
.. has_math: true

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



Day off
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Next Thurs Apr 8 is ECSE Wellness Day.   There will be no class, nor homework due.

2nd Exam
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#. Mon Apr 5, same rules as before.

#. It will cover material up thru class 16 (3/22/21).

#. That's roughly up thru Radke video 46.


New material
-------------

#. 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.



#. Important concept in the noisy channel example (with X and N both being
   Gaussian):   The most likely value of X given Y is
   not Y but is somewhat smaller, depending on the relative sizes of
   :math:`\sigma_X` and :math:`\sigma_N`.  This is true in spite of :math:`\mu_N=0`. It
   would be really useful for you to understand this intuitively.  Here's
   one way:

   If you don't know Y, then the most likely value of X is 0.  Knowing Y
   gives you more information, which you combine with your initial info
   (that X is :math:`N(0,\sigma_X)` to get a new estimate for the most likely X.
   The smaller the noise, the more valuable is Y.  If the noise is very
   small, then the mostly likely X is close to Y.  If the noise is very
   large (on average) then the most likely X is still close to 0.

   

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

In class 15, I tried to motivate the effect of changing one variable on probability density.   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.

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