Noise & Uncertainty

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(,)

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

Gaussian Distribution Uncertainty of measurement expressed in terms of σ

Gaussian Distribution : FWHM +t 

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. • Independent variable = non-negative integer number • 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 use

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? 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.

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),

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

Examples • mB=10.0±0.2 and mV=9.0±0.1 What is the uncertainty of mB-mV?

Examples • 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 Gaussian distribution Poisson distribution • Accuracy vs. precision • Probability distributions and confidence levels • Central Limit Theorem • Propagation of Errors • Weighted means • Chapter/sections covered in this lecture : 2

Noise & Uncertainty