Engineering Probability Class 13 Thurs 2021-03-11

W Randolph Franklin (WRF), Rensselaer Polytechnic Institute (RPI)

2021-03-11 00:00

Source

Table of contents::

1 Matlab

Matlab, Mathematica, and Maple all will help you do problems too big to do by hand. Sometime I'll demo one or the other.

Matlab

Major functions:
```
cdf(dist,X,A,...)
pdf(dist,X,A,...)
```

Common cases of dist (there are many others):

'Binomial'
'Exponential'
'Poisson'
'Normal'
'Geometric'
'Uniform'
'Discrete Uniform'

Examples:

pdf('Normal',-2:2,0,1)
cdf('Normal',-2:2,0,1)

p=0.2
n=10
k=0:10
bp=pdf('Binomial',k,n,p)
bar(k,bp)
grid on

bc=cdf('Binomial',k,n,p)
bar(k,bc)
grid on

x=-3:.2:3
np=pdf('Normal',x,0,1)
plot(x,np)

Interactive GUI to explore distributions: disttool

Random numbers:

rand(3)
rand(1,5)
randn(1,10)
randn(1,10)*100+500
randi(100,4)

Interactive GUI to explore random numbers: randtool

Plotting two things at once:

x=-3:.2:3
n1=pdf('Normal',x,0,1)
n2=pdf('Normal',x,0,2)
plot(x,n1,n2)
plot(x,n1,x,n2)
plot(x,n1,'--r',x,n2,'.g')

Use Matlab to compute a geometric pdf w/o using the builtin function.
Review. Which of the following do you prefer to use?
1. Matlab
2. Maple
3. Mathematica
4. Paper. It was good enough for Bernoulli and Gauss; it's good enough for me.
5. Something else (please email about it me after the class).

1.1 My opinion

This is my opinion of Matlab.

Advantages
1. Excellent quality numerical routines.
2. Free at RPI.
3. Many toolkits available.
4. Uses parallel computers and GPUs.
5. Interactive - you type commands and immediately see results.
6. No need to compile programs.
Disadvantages
1. Very expensive outside RPI.
2. Once you start using Matlab, you can't easily move away when their prices rise.
3. You must force your data structures to look like arrays.
4. Long programs must still be developed offline.
5. Hard to write in Matlab's style.
6. Programs are hard to read.
Alternatives
1. Free clones like Octave are not very good
2. The excellent math routines in Matlab are also available free in C++ librarues
3. With C++ libraries using template metaprogramming, your code looks like Matlab.
4. They compile slowly.
5. Error messages are inscrutable.
6. Executables run very quickly.

2 Chapter 4 ctd

Text 4.2 p 148 pdf
Simple continuous r.v. examples: uniform, exponential.
The exponential distribution complements the Poisson distribution. The Poisson describes the number of arrivals per unit time. The exponential describes the distribution of the times between consecutive arrivals.

The exponential is the continuous analog to the geometric. If the random variable is the integral number of seconds, use geometric. If the r.v. is the real number time, use exponential.

Ex 4.7 p 150: exponential r.v.
Properties
1. Memoryless.
2. $f(x) = \lambda e^{-\lambda x}$ if $x\ge0$, 0 otherwise.
3. Example: time for a radioactive atom to decay.
Skip 4.2.1 for now.
The most common continuous distribution is the normal distribution.
4.2.2 p 152. Conditional probabilities work the same with continuous distributions as with discrete distributions.
p 154. Gaussian r.v.
1. $f(x) = \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(x-\mu)^2}{2\sigma^2}}$
2. cdf often called $\Psi(x)$
3. cdf complement:
  1. $Q(x)=1-\Psi(x) = \int_x^\infty \frac{1}{\sqrt{2\pi} \cdot \sigma} e^{\frac{-(t-\mu)^2}{2\sigma^2}} dt$
  2. E.g., if $\mu=500, \sigma=100$,
    1. P[x>400]=0.66
    2. P[x>500]=0.5
    3. P[x>600]=0.16
    4. P[x>700]=0.02
    5. P[x>800]=0.001
Text 4.3 p 156 Expected value
Skip the other distributions (for now?).