Power of Probability
Principle of Maximum Likelihood
Weighted Averages
Linear Least Squares Fitting
Lecture # 6
Physics 2BL
Summer 2015
Principle of Maximum Likelihood
• Best estimates of X and s from N
measurements (x1 - xN) are those for which
ProbX,s (xi) is a maximum
Clicker Question 8
Upon flipping a coin three times, what are the
chances of three heads in a row?
(a)
(b)
(c)
(d)
(e)
1
0.5
0.25
0.125
0.0625
Clicker Question 8.5
What are the chances that two people in this
room have a Birthday within one day of
someone else?
(a)
(b)
(c)
(d)
(e)
> 80%
60 - 80%
40 - 60%
20 - 40%
< 20%
Yagil
Yagil
Yagil
Weighted averages (Chapter 7)
Yagil
Weighted averages
• X=x=
• swav =
where
wi =
1
s i2
Yagil
Clicker Question 9
Two measurements of the speed of sound give the answers:
uA = (332 ± 1) m/s and uB = (339 ± 3) m/s.
What is the random chance of getting two results that is  this
difference ?
(A)
(B)
(C)
(D)
(E)
2%
3%
4%
8%
40%
Clicker Question 10
Two measurements of the speed of sound give the answers:
uA = (332 ± 1) m/s and uB = (339 ± 3) m/s.
What is the best estimate (weighted mean)?
(A)
(B)
(C)
(D)
(E)
336.5 ± 2 m/s
336 ± 2 m/s
336.5 ± 0.9 m/s
332.7 ± 0.9 m/s
333 ± 2 m/s
b) Best estimate is the weighted mean:
Linear Relationships: y = A + Bx
(Chapter 8)
8
Slope = 1.01
Y value
• Data would lie on a
straight line, except
for errors
• What is ‘best’ line
through the points?
• What is uncertainty in
constants?
• How well does the
relationship describe
the data?
6
4
2
0
0
2
4
X Value
6
8
Analytical Fit
8
Slope = 1.01 ± 0.01
Y value
• Best means ‘minimize
the square of the
deviations between
line and points’
• Can use error analysis
to find constants, error
6
4
2
0
0
2
4
X Value
6
8
The Details of How to Do This
(Chapter 8)
y  A  Bx
• Want to find A, B that
y
minimize difference
 deviation
yi
of yi
between data and line

• Since line above some
yi  y  yi  A  Bx i
data, below other,
N
2
(y

A

Bx
)

minimize sum of
i
i
i1
squares of deviations 

  y i  AN  B x i  0
• Find A, B that
A

minimize this sum
2
B
  x i y i  A x i  B x i  0
Finding A and B

  y i  AN  B x i  0
A

2
  x i y i  A x i  B x i  0
B
• After minimization,
solve equations for A
and B
• Looks nasty, not so
bad…
• See Taylor, example
8.1

A
B
2
x
 i  yi   xi  xi yi

N xi yi   xi  yi

  N  xi 
2
 x 
2
i
Uncertainty in Measurements of y
N
• Before, measure
1
2
sx 
(x

x
)

i
several times and take
N i1
standard deviation as
error in y
• Can’t now, since yi’s 
N
1
2
(y

A

Bx
)
are different quantities s y 

i
i
N  2 i1
• Instead, find standard
deviation of deviations

Uncertainty in A and B
• A, B are calculated
from xi, yi
• Know error in xi, yi ;
use error propagation
to find error in A, B
• A distant extrapolation
will be subject to large
uncertainty

sA  sy
sB  sy
2
x
 i

N

  N xi 
2
 x 
2
i
Uncertainty in x
• So far, assumed
negligible uncertainty
in x
• If uncertainty in x, not
y, just switch them
• If uncertainty in both,
convert error in x to
error in y, then add
errors
equivalent
error in y
actual
error in x
y  Bx
s y (equiv)  Bs x
s y (equiv)  s y 2  Bs x 
2
Other Functions
y  Ae Bx
• Convert to linear
• Can now use least
squares fitting to get ln
A and B
ln y  ln A  Bx

Experiment 3
• Goals: Test model for damping
• Model of a shock absorber in car
• Procedure: develop and demonstrate critically
damped system
• check out setup, take data, do data make sense?
• Write up results - Does model work under all
conditions, some conditions? Need modification?
Simple Harmonic Motion
x  x0
3
dipslacement
• Spring provides
linear restoring force
 Mass on a spring
is a harmonic
oscillator
2
x 0
1
0
-1
F  kx
d2x
m 2  kx
dt
x(t)  x0 cos t

-2
-3
0
10
20
30
time
T
2

40


50
60
k

m
Damped SHM
• Consider both position
and velocity
dependant forces
• Behavior depends on
how much damping 
occurs during one
‘oscillation’


 b 
k
b 2 
.
x  x 0 exp 
t cost

2 
 2m  
 m 4m 
d 2x
dx
m 2  kx b
dt
dt
 b   k
b 2 

x  x0 exp
t exp
it

2 

 2m   m 4m 
or

22

 b
b
k
b
k

exp

 
tt
xx xx00 exp

22



 22m
4m
mm
m
4
m



Relative Damping Strength:
Weak damping
 b   k
b 2 
.
x  x 0 exp 
t cost

2 
 2m  
m
4m


dipslacement
3
b2
k

4m2
m
2
1
0
6
-1
4
position

weak damping
2
-2
0
0
-2
2
4
6
(underdamped)
-4
-6
t ime
8
10
0
100
200
300
400
time
500
600
Relative Damping Strength:
Strong damping
b2
k

4m 2
m
strong damping
3
dipslacement
2
 b

b
k
t
x  x0 exp  


2
 2m

4
m
m


2
1
0
0
(overdamped)
5
10
15
time
20
25
Relative Damping Strength:
Critical damping
b2
k

2
4m
m
critical damping

t


3
dipslacement
2
 b
k
b
x  x0 exp  


2
 2m
m
4
m

2
1
0
-100
bcrit = 2 mk
0
100 200 300 400 500 600 700
time
Comparison of the various types
of damping
dipslacement
3
overdamped
critically
damped
2
1
0
-1
underdamped
-2
-100
0
100 200 300 400 500 600 700
time
Terminal Velocity
For velocity:
.
y(t) = vt[1 - e-(b/m)t]
Experimental Setup for Falling
Mass and Drag
How do you measure velocity?
Plotting Graphs
Give each graph a title
Determine independent and dependent variables
Determine boundaries
Include error bars
Demonstrate critical damping:
show convincing evidence that
critical damping was achieved
• Demonstrate that damping is critical
– No oscillations (overshoot)
– Shortest time to return to equilibrium position
Remember
• Write-up for Experiment # 3
• Homework Taylor #8.6, 8.10
– Last assignment
• Read Taylor Chapter 12