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Engineering Probability Class 15 Thu 2020-03-05

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1   Review of normal (Gaussian) 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. Review: 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. .68
    5. .84
  4. Repeat that question for the interval [500,700].
  5. Repeat that question for the interval [0,300].

2   Varieties of Gaussian functions

  1. Book page 167: $\Phi(x)$ is the CDF of the Gaussian.
  2. Book page 168 and table on page 169: $Q(x) = 1 - \Phi(x)$.
  3. Mathematica (and other SW packages): Erf[x] is integral of pdf from 0 to x $Erf(x) = Q(x)-.5$ .
  4. Erfx(x) = 1-Erf(x).

(The nice thing about standards is that there are so many of them.)

3   Mathematica on Gaussians

  1. NormalDistribution[m,s] is the abstract pdf.
  2. get functions of it thus:
    1. PDF[NormalDistribution[m,s][x]]
    2. CDF ...
    3. Mean, Variance, Median ..
  3. MultinormalDistribution[{mu1, mu2}, {{sigma11, sigma12}, {sigma12, sigma22}}] (details later).

4   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. There's no calculation here, but this topic is used for several future problems.
  5. Example 5.5 on page 238.
  6. Example 5.6 on page 240. Easy, look at it yourself.
  7. Example 5.7 on page 241. Easy, look at it yourself.
  8. Example 5.8 on page 242. Easy, look at it yourself.
  9. Example 5.9 on page 242.
  10. 5.3 Joint CDF page 242.
  11. Example 5.11 on page 245. What is f(x,y)?
  12. Example 5.12 p 246
  13. 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]?
  14. Example 5.14 on page 247.
  15. Example 5.16 on page 252.