1 Applied electrical engineering

My house has solar cells and powerwall. As I write this (245pm Thurs), I have generated more than I've used since the start of the week (00:00 Monday). It's not even spring.

2 Chapter 3 Discrete Random Variables, 2

Starting around page 110.

1. 3.3.2 page 109 Variance of an r.v.

   1. That means, how wide is its distribution?

   2. Example: compare the performance of stocks vs bonds from year to year. The expected values (means) of the returns may not be so different. (This is debated and depends, e.g., on what period you look at). However, stocks' returns have a much larger variance than bonds.

   3. \(\sigma^2_X = VAR[X] = E[(X-m_X)^2] = \sum (x-m_x)^2 p_X(x)\)

   4. standard deviation \(\sigma_X = \sqrt{VAR[X]}\)

   5. \(VAR[X] = E[X^2] - m_X^2\)

   6. 2nd moment: \(E[X^2]\)

   7. also 3rd, 4th... moments, like a Taylor series for probability

   8. shifting the distribution: VAR[X+c] = VAR[X]

   9. scaling: \(VAR[cX] = c^2 VAR[X]\)

2. Derive variance for Bernoulli.

3. Example 3.20 3 coin tosses

   1. general rule for binomial: VAR[X]=npq

   2. Derive it.

   3. Note that it sums since the events are independent.

   4. Note that variance/mean shrinks as n grows.

4. Geometric distribution: review mean and variance.

5. Suppose that you have just sold your internet startup for $10M. You have retired and now you are trying to climb Mt Everest. You intend to keep trying until you make it. Assume that:

   1. Each attempt has a 1/3 chance of success.

   2. The attempts are independent; failure on one does not affect future attempts.

   3. Each attempt costs $70K.

   Review: What is your expected cost of a successful climb?

   1. $70K.

   2. $140K.

   3. $210K.

   4. $280K.

   5. $700K.

6. 3.4 page 111 Conditional pmf

7. Example 3.24 Residual waiting time

   1. X, time to xmit message, is uniform in 1...L.

   2. If X is over m, what's probability that remaining time is j?

   3. \(p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)\)

8. \(p_X(x) = \sum p_X(x|B_i) P[B_i]\)

9. Example 3.25 p 113 device lifetimes

   1. 2 classes of devices, geometric lifetimes.

   2. Type 1, probability \(\alpha\), parameter r. Type 2 parameter s.

   3. What's pmf of the total set of devices?

10. Example 3.26, p114.

11. 3.5 p115 More important discrete r.v

12. Table 3.1: We haven't seen \(G_X(z)\) yet.

13. 3.5.1 p 117 The Bernoulli Random Variable

    We'll do mean and variance.

14. Example 3.28 p119 Variance of a Binomial Random Variable

15. Example 3.29 Redundant Systems

16. 3.5.3 p119 The Geometric Random Variable

    It models the time between two consecutive occurrences in a sequence of independent random events. E.g., the length of a run of white bits in a scanned image (if the bits are independent).

17. 3.5.4 Poisson r.v.

    1. The experiment is observing how many of a large number of rare events happen in, say, 1 minute.

    2. E.g., how many cosmic particles hit your DRAM, how many people call to call center.

    3. The individual events are independent. (In the real world this might be false. If a black hole occurs, you're going to get a lot of cosmic particles. If the ATM network crashes, there will be a lot of calls.)

    4. The r.v. is the number that happen in that period.

    5. There is one parameter, \(\alpha\). Often this is called \(\lambda\).

       \begin{equation*} p(k) = \frac{\alpha^k}{k!}e^{-\alpha} \end{equation*}

    6. Mean and std dev are both \(\alpha\).

    7. In the real world, events might be dependent.

18. Example 3.32 p123 Errors in Optical Transmission

19. 3.5.5 p124 The Uniform Random Variable