.. title: Engineering Probability Class 9 Mon 2019-02-11
.. slug: class09
.. date: 2019-02-11
.. tags: mathjax
.. category: class
.. link: 
.. description: 
.. type: text

.. sectnum::
.. contents:: Table of contents
..

   
Chapter 3 ctd
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#. Geometric distribution: review mean and variance.

#. 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:

   a. Each attempt has a 1/3 chance of success.
   #. The attempts are independent; failure on one does not affect future attempts.
   #. Each attempt costs $70K.

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

   a. $70K.
   #. $140K.
   #. $210K.
   #. $280K.
   #. $700K.

   
#. 3.4 page 111 Conditional pmf

#. Example 3.24 Residual waiting time
   
   #. X, time to xmit message, is uniform in 1...L.
   #. If X is over m, what's probability that remaining time is j?
   #. :math:`p_X(m+j|X>m) = \frac{P[X =m+j]}{P[X>m]} = \frac{1/L}{(L-m)/L} = 1/(L-m)`

#. :math:`p_X(x) = \sum p_X(x|B_i) P[B_i]`

#. Example 3.25 p 113 device lifetimes
   
   #. 2 classes of devices, geometric lifetimes.
   #. Type 1, probability :math:`\alpha`, parameter r.  Type 2 parameter s.
   #. What's pmf of the total set of devices?

#. Example 3.26.

#. 3.5 More important discrete r.v

#. Table 3.1: We haven't seen :math:`G_X(z)` yet.

#. 3.5.4 Poisson r.v.
   
   #. The experiment is observing how many of a large number of rare events happen in, say, 1 minute.  
   #. E.g., how many cosmic particles hit your DRAM, how many people call to call center.
   #. 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.)*
   #. The r.v. is the number that happen in that period.
   #. There is one parameter, :math:`\alpha`.  Often this is called    :math:`\lambda`.

      .. math::
	 
	 p(k) = \frac{\alpha^k}{k!}e^{-\alpha}
	 
   #. Mean and std dev are both :math:`\alpha`.
   #. In the real world, events might be dependent.