Binomial Distribution Possion Distribution Gaussian Distribution Lorentzian Distribution Error Analysis

1. Noise in the data Raw data = scientific data + noise Noise : random noise, systematic noise Random noise : simple statistical relations(eg normal distribution) systematic noise : known sources or unknown sources Precision( 정밀도 ) :how well measure the results ( 그림 a) Accuracy( 정확도 ) : how close to true values ( 그림 b) Systematic noise : 정밀도가 높아도 정확도가 높지 않음 통계처리가 어렵다. CCD 의 바이아스 (bias level), 전처리 광 (preflash), 오래된 검출기 온도변화에 의한 gain 변화, 관측시 얇은 구름 등 무작위 오차 : 통계적 오차, 정밀도 개선 : 실험 방법 개선, 정밀측정기술개발, 반복 실험, 등

The number of significant figures 1. the leftmost nonzero digit is the most significant digit. 2. If there is no decimal point, the rightmost nonzero digit is the least significant digit 3. If there is a decimal point, the rightmost digit is the least significant digit, even if it is a All digit between the least and most significant digits are counted as significant digits Eg ; four digits are significant : *10 3 three digits are significant : 1010 Experiment result : number of significant digits = one more than that dictated by the experimental precision Eg : L = or L= Round off : 1. > ½, increment. 2. < ½, no increment, 3. ½,increment if odd With computer analysis, it is generally advisable to retain all available digits in intermedidate calculations and round only the final results

Random error –statistical fluctuations = noise x 을 무한 번 측정할 경우 ==> 자료 분포 == 특정 x 를 얻을 확률을 나타내는 분 포 이를 모 분포 (Parent distribution) 샘플 분포 (sample distribution) 실선 : 측정치 ( 히스토그램 ) 에서 계산한 평균 19.9, 표준편차 0.52 의 가우시안 곡선 점선 : 모분포 ( 평균 20.0, 표준편차 =0.50 변수 x 측정에서 면적 ( area) / 커브의 전면적 (total area) = 확률 P(x) dx : x 가 x+-dx 내에 있을 확률 실험과 관측에 의한 히스토그램 (histogram) 과 이론적 가우스 (Gaussian) 곡선

실험 또는 관측 분포와 모 분포

실험 또는 관측 분포와 모 분포 -cont

Summary The parent distribution is characterized by its moments: Parent probability distribution: p(x) Mean: first moment. μ = int x p(x) dx Variance: second moment. Var(x) = int (x − μ) 2 p(x) dx “Sigma”:  = sqr Var(x) Aliases:  is the standard deviation, but is also known as the “dispersion” or “rms dispersion” μ measures the “center” and  measures the “width” of the parent distribution. NB: the mean can be very different from the median (50 th percentile) or the mode (most frequent value) of the parent distribution. These represent alternative measures of the distribution’s “center.” But the mean is the more widely used parameter.

2. Probability Distribution : 1. Gaussian or normal Error Distribution : 많은 실험의 무작위 측정치들 2. Poisson Distribution ; 카운트 실험 (counting experiments - data represent the number of items or events observed per interval) 3. binomial Distribution experiments in which the results is one of a small number of possible final states ( head and tail in a series of coin tosses, or the number of particles scattered forward or backward rel to the direction of the incident particles) Poisson and Gaussian = limiting cases of the binomial distribution

2. 1 Binomial Distribution

Algebraic derivations of mean complete binomial probability mass functions are always unity the expected value of a discrete random variable to the binomial distributionexpected valuediscrete random variable, with m = n − 1 and s = k − 1

Algebraic derivations of variance, with m = n − 1 and s = k − 1

2. 2 Poisson Distribution

Poisson distribution

확률 유도

평균과 표준편차

확률 P(x,  ) =  x e -  /x!

Poisson distribution, probability

숙제 1. add three rows to previous table for values of the Poisson function for the expected means, m = 0.01, 0.1, and 0.3, each for x= 0, 1, 2, 3, plot histograms(by hand or with computer) for these values and also those for m=1 on a single plot. Use an expanded scale; truncate the plot after x=4. Comment on the behavior of the four histograms.

2. 3. Gaussian or Normal Error Distribution

표준 가우스 분포

적분 확률 (interal probability)

Normal Distribution Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue), while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.standard deviationmean

Poisson & normal distributions

2. 4. Lorentzian 분포

관측기기 Statistical, and systemetic errors. Statistical fluctuation  noise ; inherent randomness of certain types of events. Eg ; detect photons from a source : the actual number detected in a time interval will fluctuate from interval to interval  Theoretically poisson distribution 예 CCD : cosmic ray, readout, dark noise, gain change due to T changes, the aging detector or electronic components 표준편차 +- 1    의  측정치들

Types of CCD calibration data Additive systematic noise (bais, dark, fringe in nearIR) Multiplicative systematic noise (flat) Fig ; a) counts and incident light intensity for two pixels on a CCD B) additive correction to the data – zero counts for no light (bias, dark, fringe) C) correct multiplicative noise(pixel to pixel QE variation and non-uniform illumination)

a) counts and incident light intensity for two pixels on a CCD b) additive correction (bias, dark, fringe) to the data  zero counts for no light c) Multicative correction(flat and non- uniform illumination)