Engineering Probability Class 24 Mon 20180416
Table of contents
1 Material from text

Example 5.47, page 282: Estimation of signal in noise

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.

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.
 We're throwing darts uniformly at a one foot square dartboard.
 We observe 2 random variables, X, Y, where the dart hits (in Cartesian coordinates).
 $$f_{X,Y}(x,y) = \begin{cases} 1& \text{if}\,\, 0\le x\le1 \cap 0\le y\le1\\ 0&\text{otherwise} \end{cases}$$
 $$P[.5\le x\le .6 \cap .8\le y\le.9] = \int_{.5}^{.6}\int_{.8}^{.9} f_{XY}(x,y) dx \, dy = 0.01 $$
 Transform to centimeters: $$\begin{bmatrix}V\\W\end{bmatrix} = \begin{pmatrix}30&0\\0&30\end{pmatrix} \begin{bmatrix}X\\Y\end{bmatrix}$$
 $$f_{V,W}(v,w) = \begin{cases} 1/900& \text{if } 0\le v\le30 \cap 0\le w\le30\\ 0&\text{otherwise} \end{cases}$$
 $$P[15\le v\le 18 \cap 24\le w\le27] = \int_{15}^{18}\int_{24}^{27} f_{VW}(v,w)\, dv\, dw = \frac{ (1815)(2724) }{900} = 0.01$$
 See Section 5.8.3 on page 286.
3 Chapter 6: Vector random variables
 Skip the starred sections.
 Examples:
 arrivals in a multiport switch,
 audio signal at different times.
 pmf, cdf, marginal pmf and cdf are obvious.
 conditional pmf has a nice chaining rule.
 For continuous random variables, the pdf, cdf, conditional pdf etc are all obvious.
 Independence is obvious.
 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.