Engineering Probability Class 9 Mon 2019-02-11

1   Chapter 3 ctd

1. Geometric distribution: review mean and variance.

2. 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.

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

   1. $70K.
   2. $140K.
   3. $210K.
   4. $280K.
   5. $700K.

3. 3.4 page 111 Conditional pmf

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

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

6. 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?

7. Example 3.26.

8. 3.5 More important discrete r.v

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

10. 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.