Noise & Uncertainty ASTR 3010 Lecture 7 Chapter 2.
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Transcript of Noise & Uncertainty ASTR 3010 Lecture 7 Chapter 2.
Probability Distribution : P(x)
• Uniform, Binomial, Maxwell, Lorenztian, etc…• Gaussian Distribution = continuous probability distribution which describes
most statistical data well N(,)
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mean: P(x)⋅ x dx = μ−∞
∞
∫variance : P(x)⋅ (x − μ)2 dx = μ
−∞
∞
∫ =σ 2
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
Normalized…
mean
when
€
P x;n,ρ( ) =n!
x!(n − x)!ρ x (1− ρ )n−x
€
P x;n,ρ( )x=0
n
∑ =1
€
P x;n,ρ( )x=0
n
∑ ⋅ x =K = np
€
n →∞ ⇒ Normaldistribution
n →∞ and np = const ⇒ Poissonian distribution
Gaussian Distribution
€
G(x) =1
2πσ 2exp −
x − μ( )2
2σ 2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Uncertainty of measurement expressed in terms of σ
Central Limit Theorem
• Sufficiently large number of independent random variables can be approximated by a Gaussian Distribution.
Poisson Distribution
• Describes a population in counting experiments number of events counted in a unit time.o Independent variable = non-negative integer numbero Discrete function with a single parameter μprobability 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)
€
PP (x;μ) =μ x
x!e−μ
Poisson distribution
Mean and Variance
€
x = xx=0
∞
∑ PP (x;μ) = xμ x
x!e−μ
⎛
⎝ ⎜
⎞
⎠ ⎟
x=0
∞
∑
=K
= μ
(x − μ)2 =K
= x 2 − μ 2
=K
= μ €
μx
x!x=0
∞
∑ = eμuse
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) = 10S/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?
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.
2
2
1 22
1
11
c
ii
n
ii
c
n
iiic wwxx
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
23
2
3
22
2
2
21
2
1
2 ... nn
F x
F
x
F
x
F
x
F
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?