Engineering Probability Class 23 Thu 2018-04-12

Source

Table of contents

1   Material from text

  1. Example 5.35 Maximum A Posteriori Receiver on page 268.

  2. Example 5.37, page 270.

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

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

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

  5. 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.

  6. Example 5.39 Sum of Two Random Variables, page 271.

  7. 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, $\sqrt{2}$ ).

  8. Example 5.44, page 275. Tranform two independent Gaussian r.v from

    (X,Y) to (R, $\theta$).

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

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

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

    2. The r.v. are probably dependent. $\rho$} says how much.

    3. 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.

    4. The lines of equal probability density are ellipses.

    5. The marginal pdf is a 1 variable Gaussian.