Engineering Probability Class 9 Thurs 2018-02-15

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2   Chapter 3 ctd

  1. Example 3.22 Variance of geometric r.v. We'll derive it.

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

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

  4. 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?

  5. Example 3.26.

  6. 3.5 More important discrete r.v

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

  8. 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.

    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.