Noise & Uncertainty ASTR 3010 Lecture 7 Chapter 2

Accuracy & Precision

True value systematic error

Probability Distribution : P(x) Uniform, Binomial, Maxwell, Lorenztian, etc… Uniform, Binomial, Maxwell, Lorenztian, etc… Gaussian Distribution = continuous probability distribution which describes most statistical data well  N( ,  ) Gaussian Distribution = continuous probability distribution which describes most statistical data well  N( ,  )

Binomial Distribution Two outcomes : ‘success’ or ‘failure’ Two outcomes : ‘success’ or ‘failure’ probability of x successes in n trials with the probability of a success at each trial being ρ Normalized…meanwhen

Gaussian Distribution Uncertainty of measurement expressed in terms of σ

Gaussian Distribution : FWHM +t+t 

Central Limit Theorem Sufficiently large number of independent random variables can be approximated by a Gaussian Distribution. Sufficiently large number of independent random variables can be approximated by a Gaussian Distribution.

Poisson Distribution Describes a population in counting experiments Describes a population in counting experiments  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) at time of t, the probability of measuring x raindrops = P(x, 3.25)

Poisson distribution Mean and Variance use

Signal to Noise Ratio S/N = SNR = Measurement / Uncertainty S/N = SNR = Measurement / Uncertainty In astronomy (e.g., photon counting experiments), In astronomy (e.g., photon counting experiments), uncertainty = sqrt(measurement)  Poisson statistics uncertainty = sqrt(measurement)  Poisson statisticsExamples: 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? 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 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. 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. 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? w i = (11.1, 2.0, 25.0) x c = … = 8.15 kpc  c = 0.16 kpc So the best estimate is 8.15±0.16 kpc.

Propagation of Uncertainty You took two flux measurements of the same object. You took two flux measurements of the same object. F 1 ±  1, F 2 ±  2 Your average measurement is F avg =(F 1 +F 2 )/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 m avg and its uncertainty?  F  ?   m You need to express above flux measurements in magnitude (m = 2.5log(F)). Then, what’s m avg and its uncertainty?  F  ?   m For a function of n variables, F=F(x1,x2,x3, …, x n ), For a function of n variables, F=F(x1,x2,x3, …, x n ),

Examples 1.S=1/2bh, b=5.0±0.1 cm and h=10.0±0.3 cm. What is the uncertainty of S? What is the uncertainty of S? S S h b

Examples 2.m B =10.0±0.2 and m V =9.0±0.1 What is the uncertainty of m B -m V ? What is the uncertainty of m B -m V ?

Examples 3.M = m - 5logd + 5, and d = 1/π = 1000/π HIP m V =9.0±0.1 mag and π HIP =5.0±1.0 mas. What is M V and its uncertainty?

In summary… Important Concepts Accuracy vs. precision Probability distributions and confidence levels Central Limit Theorem Propagation of Errors Weighted means Important Terms Gaussian distribution Poisson distribution Chapter/sections covered in this lecture : 2