Engineering Probability Class 23 Thu 2018-04-12
Table of contents
1 Material from text
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Example 5.35 Maximum A Posteriori Receiver on page 268.
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Example 5.37, page 270.
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Remember equations 5.49 a,b for total probability on page 269-70 for conditional expectation of Y given X.
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Section 5.8 page 271: Functions of two random variables.
- This is an important topic.
- Linear transformation of two Gaussian r.v.
- Sum and difference of 2 Gaussian r.v. are independent.
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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.
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Example 5.39 Sum of Two Random Variables, page 271.
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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
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(X,Y) to (R, $\theta$).
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Section 5.9, page 278: pairs of jointly Gaussian r.v.
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I will simplify formula 5.61a by assuming that $\mu=0, \sigma=1$.
$$f_{XY}(x,y)= \frac{1}{2\pi \sqrt{1-\rho^2}} e^{ \frac{-\left( x^2-2\rho x y + y^2\right)}{2(1-\rho^2)} } $$ .
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The r.v. are probably dependent. $\rho$} says how much.
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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.
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The lines of equal probability density are ellipses.
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The marginal pdf is a 1 variable Gaussian.
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