Noise & Uncertainty
ASTR 3010
Lecture 7
Chapter 2
Accuracy & Precision
Accuracy & Precision
True value
systematic
error
Probability Distribution : P(x)
• Uniform, Binomial, Maxwell, Lorenztian, etc…
• Gaussian Distribution = continuous probability distribution which describes
most statistical data well  N(,)

variance: 
m ean:



P(x) x dx  
2
2
P(x)
(x


)
dx





Binomial Distribution
• Two outcomes : ‘success’ or ‘failure’
probability of x successes in n trials with the probability of a success at each trial
being ρ
n!
P x;n,  
 x (1   ) n  x
x!(n  x)!
Normalized…
n

 Px;n,   1
x 0
mean
n

 Px;n,  x  K
 np
x 0
when

n 
 Norm aldistribution
n  and np  const  Poissoniandistribution
Gaussian Distribution
 x   2 


1


G(x) 
exp

2

2 2
 2 


Uncertainty of measurement expressed in terms of σ
Gaussian Distribution : FWHM
1
G(  t)  G()  t 1.177
2
2.355


+t
Central Limit Theorem
• Sufficiently large number of independent random variables can be
approximated by a Gaussian Distribution.
Poisson Distribution
PP (x; ) 
• Describes a population in counting experiments
x
x!
e 
 number of events counted in a unit time.
o Independent variable = non-negative integer number
o Discrete function with a single parameter
 μ
probability of seeing x events when the average event rate is 
E.g., average number of raindrops per second for a storm = 3.25 drops/sec
at time of t, the probability of measuring x raindrops = P(x, 3.25)
Poisson distribution
Mean and Variance
  x   
x   xPP (x; )  x e 

x 0
x 0  x!


K

(x  )
2

K
 x
use
2

K

 x!  e 
x 0
2

x
Signal to Noise Ratio
• S/N = SNR = Measurement / Uncertainty
• In astronomy (e.g., photon counting experiments),
uncertainty = sqrt(measurement)  Poisson statistics
Examples:
• From a 10 minutes exposure, your object was detected at a signal strength
of 100 counts. Assuming there is no other noise source, what is the S/N?
S = 100  N = sqrt(S) = 10
S/N = 10 (or 10% precision measurement)
• For the same object, how long do you need to integrate photons to achieve
1% precision measurement?
For a 1% measurement, S/sqrt(S)=100  S=10,000. Since it took 10
minutes to accumulate 100 counts, it will take 1000 minutes to achieve
S=10,000 counts.
Weighted Mean
• Suppose there are three different measurements for the distance to the
center of our Galaxy; 8.0±0.3, 7.8±0.7, and 8.25±0.20 kpc.
What is the best combined estimate of the distance and its uncertainty?
n
xc   xi wi
i 1
1
 c2  i 1 1 2
i
n
wi = (11.1, 2.0, 25.0)
xc = … = 8.15 kpc
c= 0.16 kpc
So the best estimate is 8.15±0.16 kpc.
 i2
wi  2
c
Propagation of Uncertainty
• You took two flux measurements of the same object.
F1 ±1, F2 ±2
Your average measurement is Favg=(F1+F2)/2 or the weighted mean.
Then, what’s the uncertainty of the flux?  we already know how to do this…
• You need to express above flux measurements in magnitude (m =
2.5log(F)). Then, what’s mavg and its uncertainty? F?m
• For a function of n variables, F=F(x1,x2,x3, …, xn),
2
2
2
2
 F  2
 F 
 F  2  F  2
  3  ...  
  n
  2  
 F2     12  
 x1 
 x2 
 x3 
 xn 
Examples
1.
S=1/2bh, b=5.0±0.1 cm and h=10.0±0.3 cm.
What is the uncertainty of S?
h
S
b
Examples
2.
mB=10.0±0.2 and mV=9.0±0.1
What is the uncertainty of mB-mV?
Examples
3.
M = m - 5logd + 5, and d = 1/π = 1000/πHIP
mV=9.0±0.1 mag and πHIP=5.0±1.0 mas.
What is MV and its uncertainty?
In summary…
Important Concepts
Important Terms
• Accuracy vs. precision
• Probability distributions and
confidence levels
• Central Limit Theorem
• Propagation of Errors
• Weighted means
• Gaussian distribution
• Poisson distribution
Chapter/sections covered in this lecture : 2