.. title: Engineering Probability Class 23 Thu 2018-04-12
.. slug: class23
.. date: 2018-04-12
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Material from text
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#. Example 5.35 Maximum A Posteriori Receiver on page 268.

#. Example 5.37, page 270.

#. Remember equations 5.49 a,b for total probability on page 269-70 for conditional expectation of Y given X.

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

   a. This is an important topic.
   #. Linear transformation of two Gaussian r.v.  
   #. Sum and difference of 2 Gaussian r.v. are independent.


#. 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 or compress Y with greater precision.  So, we want to do probabilities on all this.
      
#. Example 5.39 Sum of Two Random Variables, page 271.

#. Example 5.40 Sum of Nonindependent Gaussian Random Variables, page 272.

   I'll do an easier case of independent N(0,1) r.v.   The sum will be N(0, $\\sqrt{2}$ ).
         
#. Example 5.44, page 275. Tranform two independent Gaussian r.v from
      (X,Y) to (R, $\\theta$).  



#. 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.