.. title: Engineering Probability Class 25 Thurs 2020-04-23
.. slug: class25
.. date: 2020-04-22
.. tags: class
.. link: 
.. description: 
.. type: text
.. has_math: true

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

My opinion of Matlab
--------------------

(copied from class 9).

#. Advantages

   #. Excellent quality numerical routines.
   #. Free at RPI.
   #. Many toolkits available.
   #. Uses parallel computers and GPUs.
   #. Interactive - you type commands and immediately see results.
   #. No need to compile programs.

#. Disadvantages

   #. Very expensive outside RPI.
   #. Once you start using Matlab, you can't easily move away when their prices rise.
   #. You must force your data structures to look like arrays.
   #. Long programs must still be developed offline.
   #. Hard to write in Matlab's style.
   #. Programs are hard to read.

#. Alternatives

   #. Free clones like Octave are not very good
   #. The excellent math routines in Matlab are also available free in C++ librarues
   #. With C++ libraries using template metaprogramming, your code looks like Matlab.
   #. They compile slowly.
   #. Error messages are inscrutable.
   #. Executables run very quickly.

      
Homework grading rules
---------------------------

#. Each homework grade will be normalized to be out of 100.
#. Then the lowest homework grade will be dropped.
#. There will be only 9 homeworks.


Useful tables in book
---------------------

#. TABLE 3.1 Discrete random variables is page 115.
#. TABLE 4.1 Continuous random variables is page 164.


Worked out problems from book
-----------------------------

#. 5.30, page 263

#. 5.33, p 267

Review questions
----------------

#. What is $$\\int_{-\\infty}^\\infty e^{\\big(-\\frac{x^2}{2}\\big)} dx$$?

   a. 1/2
   #. 1
   #. $2\\pi$
   #. $\\sqrt{2\\pi}$
   #. $1/\\sqrt{2\\pi}$

#. What is the largest possible value for a correlation coefficient?

   a. 1/2
   #. 1
   #. $2\\pi$
   #. $\\sqrt{2\\pi}$
   #. $1/\\sqrt{2\\pi}$

#. The most reasonable probability distribution for the number of defects on an integrated circuit caused by dust particles, cosmic rays, etc, is

   a. Exponential
   #. Poisson
   #. Normal
   #. Uniform
   #. Binomial
      
#. The most reasonable probability distribution for the time until the next request hits your web server is:

   a. Exponential
   #. Poisson
   #. Normal
   #. Uniform
   #. Binomial

#. If you add two independent normal random variables, each with variance 10, what is the variance of the sum?

   a. 1
   #. $\\sqrt2$
   #. 10
   #. $10\\sqrt2$
   #. 20


#.  X and Y are two uniform r.v. on the interval [0,1].  X and Y are independent.  Z=X+Y.  What is E[Z]?

    a. 0
    #. 1/2
    #. 2/3
    #. 1.
    #. 2.
       
#. Now let W=max(X,Y).    What is E[W]?

   a. 0
   #. 1/2
   #. 2/3
   #. 1.
   #. 2.


#. Experiment: toss two fair coins, one after the other.   Observe two random variables:

   a. X is the number of heads.
   #. Y is the toss when the first head occurred, with 0 meaning both coins were tails.

   What is P[X=1]?

   a. 0
   #. 1/4
   #. 1/2
   #. 3/4
   #. 1

#. What is P[Y=1]?
   
   a. 0
   #. 1/4
   #. 1/2
   #. 3/4
   #. 1

#. What is P[Y=1 & X=1]?
   
   a. 0
   #. 1/4
   #. 1/2
   #. 3/4
   #. 1
      
#. What is P[Y=1|X=1]?
   
   a. 0
   #. 1/4
   #. 1/2
   #. 3/4
   #. 1
      
#. What is P[X=1|Y=1]?
   
   a. 0
   #. 1/4
   #. 1/2
   #. 3/4
   #. 1

#. These next few questions concern math SAT scores, which we assume have a mean of 500 and standard deviation of 100.

   What is the probability that one particular score is between 400 and 600?

    A. .34
    #. .68
    #. .96
    #. .98
    #. .9974

#. I take a random sample of 4 students, and compute the mean of their 4 scores.
   What is the probability that that mean is between 400 and 600?

   A. .34
   #. .68
   #. .96
   #. .98
   #. .9974

#. I take a random sample of 9 students, and compute the mean of their 9 scores.
   What is the probability that that mean is between 400 and 600?

   A. .34
   #. .68
   #. .96
   #. .98
   #. .9974

#. What is the standard deviation of the 4-student sample?
   
   A. 25
   #. 33
   #. 50
   #. 100
   #. 200
      
#. What is the standard deviation of the 9-student sample?

   A. 25
   #. 33
   #. 50
   #. 100
   #. 200