Engineering Probability Class 16 Mon 2018-03-19

Source

1   Review of normal distribution

  1. Review of the normal distribution. If $\mu=0, \sigma=1$ (to keep it simple), then: $$f_N(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} $$
  2. Show that $\int_{-\infty}^{\infty} f(x) dx =1$. This is example 4.21 on page 168.
  3. 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.
    1. .02
    2. .16
    3. .48
    4. .58
    5. .84
  4. Iclicker. Repeat that question for the interval [500,700].
  5. Iclicker. Repeat that question for the interval [0,300].

2   Chapter 5, Two Random Variables

  1. See intro I did in last class.
  2. Today's reading: Chapter 5, page 233-242.
  3. 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.
  4. Example 5.3 on page 235.
  5. Example 5.5 on page 238.
  6. Example 5.6 on page 240.
  7. Example 5.7 on page 241.
  8. Example 5.8 on page 242.

3   Next time

  1. 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]?

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

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

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

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

  5. 5.6.2 Joint moments etc

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

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

  7. Covariance, correlation coefficient.

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

    1. Discrete: Work out an example with a pair of 3-sided loaded dice.
    2. 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.

  9. 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:

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

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

    1. Increase the transmitted signal,
    2. Reduce the noise,
    3. Retransmit several times and vote.
    4. Handshake: Include a checksum and ask for retransmission if it fails.
    5. 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.

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

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

  11. 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.

  12. Functions of 2 random variables

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

  13. 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.

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

    1. 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.

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