Engineering Probability Class 24 Mon 2018-04-16

Source

1   Material from text

  1. 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.

    2. Important concept in the noisy channel example (with X and N both being Gaussian): The most likely value of X given Y is not Y but is somewhat smaller, depending on the relative sizes of \(\sigma_X\) and \(\sigma_N\). This is true in spite of \(\mu_N=0\). It would be really useful for you to understand this intuitively. Here's one way:

      If you don't know Y, then the most likely value of X is 0. Knowing Y gives you more information, which you combine with your initial info (that X is \(N(0,\sigma_X)\) to get a new estimate for the most likely X. The smaller the noise, the more valuable is Y. If the noise is very small, then the mostly likely X is close to Y. If the noise is very large (on average) then the most likely X is still close to 0.

2   Tutorial on probability density - 2 variables

In class 15, I tried to motivate the effect of changing one variable on probability density. Here's a try at motivating changing 2 variables.

  1. We're throwing darts uniformly at a one foot square dartboard.
  2. We observe 2 random variables, X, Y, where the dart hits (in Cartesian coordinates).
  3. $$f_{X,Y}(x,y) = \begin{cases} 1& \text{if}\,\, 0\le x\le1 \cap 0\le y\le1\\ 0&\text{otherwise} \end{cases}$$
  4. $$P[.5\le x\le .6 \cap .8\le y\le.9] = \int_{.5}^{.6}\int_{.8}^{.9} f_{XY}(x,y) dx \, dy = 0.01 $$
  5. Transform to centimeters: $$\begin{bmatrix}V\\W\end{bmatrix} = \begin{pmatrix}30&0\\0&30\end{pmatrix} \begin{bmatrix}X\\Y\end{bmatrix}$$
  6. $$f_{V,W}(v,w) = \begin{cases} 1/900& \text{if } 0\le v\le30 \cap 0\le w\le30\\ 0&\text{otherwise} \end{cases}$$
  7. $$P[15\le v\le 18 \cap 24\le w\le27] = \int_{15}^{18}\int_{24}^{27} f_{VW}(v,w)\, dv\, dw = \frac{ (18-15)(27-24) }{900} = 0.01$$
  8. See Section 5.8.3 on page 286.

3   Chapter 6: Vector random variables

  1. Skip the starred sections.
  2. Examples:
    1. arrivals in a multiport switch,
    2. audio signal at different times.
  3. pmf, cdf, marginal pmf and cdf are obvious.
  4. conditional pmf has a nice chaining rule.
  5. For continuous random variables, the pdf, cdf, conditional pdf etc are all obvious.
  6. Independence is obvious.
  7. Work out example 6.5, page 306. The input ports are a distraction. This problem reduces to a multinomial probability where N is itself a random variable.