.. title: Engineering Probability Class 12 Mon 2020-02-24
.. slug: class12
.. date: 2020-02-24
.. tags: class
.. link: 
.. description: 
.. type: text
.. has_math: true

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



Tutorial on probability density
-------------------------------

Since the meaning of probability density when you transform variables is
still causing problems for some people, think of changing units from
English to metric.  First, with one variable, X.

#. Let X be in feet and be U[0,1]. 

   $$f_X(x) =  \\begin{cases} 1& \\text{if }  0\\le x\\le1\\\\ 0&\\text{otherwise} \\end{cases}$$
#. $P[.5\\le x\\le .51] = 0.01$.
#. Now change to centimeters.  The transformation is $Y=30X$.
#. $$f_Y(y) =  \\begin{cases} 1/30 & \\text{if }  0\\le y\\le30\\\\ 0&\\text{otherwise} \\end{cases}$$
#. Why is 1/30 reasonable?
#. First, the pdf has to integrate to 1:  $$\\int_{-\\infty}^\\infty f_Y(y) =1$$
#. Second, $$\\begin{align} & P[.5\\le x\\le .51]  \\\\ &= \\int_.5^.51 f_X(x) dx  \\\\& =0.01  \\\\& =  P[15\\le y\\le 15.3]  \\\\& = \\int_{15}^{15.3} f_Y(y) dy  \\end{align}$$



Mathematica demo
----------------

#. Int
#. Sum
#. Manipulate
#. Binomial
   etc

Examples
--------

4.11, p153.

4.3.2 Variance
--------------

p160

   
Memoryless Exponential Distn
----------------------------

p 166.

4.4.3 Normal (Gaussian) dist
----------------------------

p 167.

Show that the pdf integrates to 1.

Lots of different notations:

Generally, F(x) = P(X<=x).

For normal:  that is called $\\Psi(x)$ .

$Q(x) = 1-\\Psi(x)$ .

Example 4.22 page 169.

4.4.4 Gamma r.v.
----------------

#. 2 parameters

#. Has several useful special cases, e.g., chi-squared and m-Erlang.

#. The sum of m exponential r.v. has the m-Erlang dist.   

   
#. Example 4.24 page 172.




Functions of a r.v.
-------------------

#. Example 4.29 page 175.

#. Linear function:  Example 4.31 on page 176.

      

	
Comic
----------

`Dilbert <../files/comics/dilbert-random.png>`_