1. Noise & Uncertainty ASTR 3010 Lecture 7 Chapter 2

2. Accuracy & Precision

3. True value systematic error

4. 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( ,  )

5. 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

6. Gaussian Distribution Uncertainty of measurement expressed in terms of σ

7. Gaussian Distribution : FWHM +t+t 

8. 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.

9. 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)

10. Poisson distribution Mean and Variance use

11. 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.

12. 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.

13. 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 ),

14. 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

15. 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 ?

16. 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?

17. 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