Transcript Steven Spielberg

Physics 114: Lecture 8
PDFs Part Deux
John F. Federici
NJIT Physics Department
Physics Cartoons
Binomial & Poisson Distributions

The binomial distribution is PB ( x; n, p) 

The mean is   np.

n!
p x (1  p) n  x .
x !(n  x)!
MatLAB: binopdf(x,n,p)
Use for yes/no statistics
The standard deviation is   np(1  p)

The Poisson distribution is PP ( x;  ) 

The mean is

The standard deviation is    .
x  .
x
x!
e  .
MatLAB: poisspdf(x,)
Use for counting statistics


SNR  
 


Example 2.3 – Quick Review

Some students measure some background counts of cosmic rays. They
recorded numbers of counts in their detector for a series of 100 2-s
intervals, and found a mean of 1.69 counts/interval. They can use the
standard deviation formula from chapter 1, which is
1
s 2   ( xi  x ) 2 ,
N
to get a standard deviation directly from the data. They do this and get s =
1.29. They can also estimate the standard deviation by   1.69  1.30.
 Now they change the length of time they count from 2-s intervals to 15-s
intervals. Now the mean number of counts in each interval will increase.
Now they measure a mean of 11.48, which implies   11.48  3.17, while
they again calculate s directly from their measurements to find s = 3.39.
 We can plot the theoretical distributions using MatLAB poisspdf(x,mu),
e.g. poisspdf(0:8,1.69) gives
ans = 0.1845 0.3118 0.2635 0.1484 0.0627 0.0212 0.0060 0.0014 0.0003
Important Point!
One can specify the UNCERTAINTY/ ERROR in a measurement
using the Standard Deviation.
For two second interval:
Measured value is 1.69±1.30 counts/second
Mean Stan. Dev
For 15 second interval:
Measured value is 11.48 ±3.17 counts/second.
We can also specify the Signal-to-Noise Ratio (SNR) as Mean
divided by Standard Deviation.
1.69
2 sec
SNR 
 1.3
1.30
Interval
15 sec SNR  11.48  3.62
3.17
Interval
Example 2.3, cont’d
Probability of counts per interval
Poisson Distribution for mean 1.69
0.3
0.25

0.2
0.15
0.1

0.05
0
0
2
4
6
8
Number of Counts
Probability of counts per interval
Poisson Distribution for mean 11.48
The plots of the distributions is shown
for these two cases in the plots at right.
 You can see that for a small mean, the
distribution is quite asymmetrical. As
the mean increases, the distribution
becomes somewhat more symmetrical
(but is still not symmetrical at 11.48
counts/interval).
 The origin of the asymmetry for SMALL
particle counts is not surprising… There
is a HARD boundary at ZERO counts.
We can not have ‘negative’ number of
counts!

0.35
0.12
0.1
0.08

0.06
0.04

0.02
0
0
5
10
15
Number of Counts per 15 s
20
25
Example 2.3, cont’d
Note as the TIME interval for the counts increases, that the AVERAGE
moves to a larger number.
 Here is the higher-mean plot with the equivalent Gaussian (normal
distribution) overlaid.

Poisson Distribution for mean 11.48
Probability of counts per interval
0.15
0.1
0.05
0
0
5
10
15
20
25
Number of Counts per 15 s

For large means (high counts), the Poisson distribution approaches the
Gaussian distribution, which we will now describe further.
Gaussian or Normal Distribution

We will simply give the expression for the Gaussian distribution without
derivation. Note that it is the limiting case of the Poisson distribution
(counting statistics) as the mean  becomes large.
 1  x   2 
1
PG ( x;  ,  ) 
exp   
 .
 2
 2    

Unlike the binomial and Poisson distributions, which are defined only for
integer values, the Gaussian distribution is continuous. That means it is a
probability density function (pdf), and to get the probability that a value will
fall between two values of x, you have to multiply by the bin width dx, or in
the limit of infinitesimal bin widths, integrate:
x2
P( x1  x  x2 )   PG ( x;  ,  )dx.
x1

Here, the mean  and standard deviation  are part of the definition of the
distribution, so are not defined separately in terms of other parameters.
Characteristics of the Gaussian
As always, the Gaussian pdf is normalized so that the area under the curve
is unity (i.e. the integral from  to  is 1). The exponential itself has unit
amplitude at x =  (i.e. exp(0) = 1), and if you do the integral of the
exponential you will find that it is  2 . Therefore, you have to divide by
this factor to normalize the integral.
 As you did in the first homework, you can determine the full width at half
maximum for the Gaussian distribution by finding where the function falls to
½ its amplitude:
 1  x 2  1
exp      
 2     2

2 x   2 2 ln 2.

A good way to think about the standard deviation is that “most” values will
lie within 1 of the mean. The actual percentage for 1 is 65%. If you go
to 2, it is 95%. If you go to 3, it is 99.7%.
Example from Exam
State which PDF (Binomial, Poisson or Gaussian) best describes the following experiments
and WHY.
a) Measuring the weight of ball bearings in a manufacturing process.
b) Measuring occurrence of defective phones on the NJIT campus.
c) Measuring the height length of a specific table in the Physics Department teaching lab.
d) Recording cosmic ray detections
e) Determining the number of students who slip and fall on campus during a snow storm.
Let’s go through each one…..
Example from Exam
State which PDF (Binomial, Poisson or Gaussian) best describes the following experiments
and WHY.
a) Measuring the weight of ball bearings in a manufacturing process.
THINK ABOUT:
• Is what you are measuring a CONTINUOUSLY varying quantity or is it DISCRETELY
varying?
GAUSSIAN
• If DISCRETELY varying, is measured quantity an INTEGER because you of
COUNTING? POISSON
• If DISCRETELY varying, is measured quantity BINARY in the sense the measured
quantity is a YES or NO?
BIONOMIAL
Example from Exam
State which PDF (Binomial, Poisson or Gaussian) best describes the following experiments
and WHY.
a) Measuring the weight of ball bearings in a manufacturing process. GAUSSIAN
b) Measuring occurrence of defective phones on the NJIT campus.
BIONOMIAL
c) Measuring the height length of a specific table in the Physics Department teaching lab.
GAUSSIAN
d) Recording cosmic ray detections
POISSON
e) Determining the number of students who slip and fall on campus during a snow storm.
BIONOMIAL
OF COURSE, since Gaussian and Poisson distributions are derived from Bionomial in
various limits, one also must consider HOW MANY times an event occurs. For example,
hopefully, the number of students falling in the snow is a SMALL number, so Bionomial is the
correct answer. For cosmic rays, if the observation time is not too large, POISSON is the
only correct answer. However, if OBERVATION TIME IS LARGE, then Gaussian will ALSO
work because the measured COUNTS becomes a more ‘continuous’ number
Example 2.3 – Prior Slide
Note as the TIME interval for the counts increases, that the AVERAGE
moves to a larger number.
 Here is the higher-mean plot with the equivalent Gaussian (normal
distribution) overlaid.

Poisson Distribution for mean 11.48
Probability of counts per interval
0.15
0.1
0.05
0
0
5
10
15
20
25
Number of Counts per 15 s

For large means (high counts), the Poisson distribution approaches the
Gaussian distribution, which we will now describe further.
Other Distributions
There are many other distributions that are met with in various
circumstances, some phenomenological, and some based on theory. An
interesting one is the Lorentzian distribution (or Cauchy distribution), which
describes the shape of spectral lines in gases or plasmas such as the Sun.
1
/2
PL ( x;  , ) 
.
  x   2    / 2 2
 This has a middle part that looks a little like a gaussian (the so-called line
core), but the parts far from the mean (called the line wings) decrease
slowly.
0.4
Gaussian ( =1)
0.35
 Although the mean is , the standard
Lorentzian (=2.354)
0.3
deviation of this distribution is undefined
0.25
(i.e. the second moment is undefined)
0.2
because of how slowly the distribution
0.15
falls off (its integral is infinite).
0.1
Normalized distributions

0.05
0
0
1
2
3
4
5
x
6
7
8
9
10
Other Distributions – 1/f noise
Depending on the type of noise present in your physical system, there are other
distribution functions can accurately predict the noise.
Voltage
Time
Examples of its occurrence include fluctuations in tide and river heights, quasar
light emissions, heart beat, firings of single neurons, and resistivity in solid-state
electronics. Wikipedia
Other Distributions – 1/f noise
Depending on the type of noise present in your physical system, there are other
distribution functions can accurately predict the noise.
White Noise
Examples of its occurrence include fluctuations in tide and river heights, quasar
light emissions, heart beat, firings of single neurons, and resistivity in solid-state
electronics. Wikipedia
Pop Culture Trivia
What classic 1982 Horror film by Steven Spielburg featured a
scene with a young child staring at ‘White Noise’ on a television
screen with the catch phrase “THEY’RE HERE”?
(a) The Shining
(b) Poltergeist
(c) An American Werewolf in London
(d) Gremlins
(e) Aliens
Pop Culture Trivia
What classic 1982 Horror film by Steven Spielberg featured a
scene with a young child staring at ‘White Noise’ on a television
screen with the catch phrase “THEY’RE HERE”?
(a)
(b) Poltergeist
(c)
(d)
(e)
In class exercise
In this exercise, we will practice analyzing images in the presence of noise. We will
continue this exercise on Thursday, so SAVE whatever you do.
On the course webpage for this week, there are data files corresponding to images
taken of a bright light source (laser) at different exposures.
Analyze…..
We interrupt this class exercise for the special announcement….
THURSDAY CLASS – Extended in-class Matlab project. Answer
any student questions in Preparation for Exam 1.
Tuesday’s class – HW5 due AND begin week 7 material
THURSDAY FEB 23rd – EXAM 1 –in class
For remainder of class time,
Start working on HW#5.