Statistics, Probability Distributions & Error Propagation. James R. Graham

Transcription

1 Statistics, Probability Distributions & Error Propagation James R. Graham

2 Sample & Parent Populations Make measurements x x In general do not expect x = x But as you take more and more measurements a pattern emerges in this sample With an infinite sample x i, i { } Expect a pattern emerges with a characteristic value Exactly specify the distribution of x i This hypothetical pool of all possible measurements is the parent population Any finite sequence is the sample population

3 Histograms & Distributions Histogram represents the frequency of discrete measurements True parent population (dotted) Inferred parent distribution (solid) 3

4 otation Parent distribution: Greek, e.g., µ Sample distribution: Latin, To determine properties of the parent distribution we assume that the properties of the sample distribution tend to those of the parent as tends to infinity x 4

5 Summation If we make measurements, x, x, x 3, etc. the sum of these measurements is x i = x + x + x x i= Typically, we use the shorthand x i = i= x i 5

6 Mean The mean of an experimental distribution is x = x i The mean of the parent population is defined as µ = lim x i 6

7 Median The median of the parent population µ / is the value for which half of x i < µ / P(x i < µ / ) = P(x i µ / ) = / The median cuts the area under the probability distribution in half 7

8 Mode The mode is the most probable value drawn from the parent distribution The mode is the most likely value to occur in an experiment For a symmetrical distribution the mean, median and mode are all the same 8

9 Deviations The deviation, d i, of a measurement, x i, from the mean, µ, is defined as d i = x i µ If µ is the true mean value the deviation is the error in x i 9

10 Mean Deviation The mean deviation vanishes Evident from the definition lim d = lim (x i µ) = lim x i µ µ 0

11 Mean Square Deviation The mean square deviation is easy to use analytically and justified theoretically σ = lim ( x i µ ) σ is also known as the variance Derive this expression = lim Computation of σ assumes we know µ x i µ

12 Population Mean Square Deviation The standard deviation, s, of a sample population is given by s = ( x i x ) The factor (-) rather than accounts for the fact that the mean was derived from the data

13 Significance The mean of the sample is the best estimate of the mean of the parent distribution The standard deviation, s, is characteristic of the uncertainties associated with attempts to measure µ But what is the uncertainty in µ? To answer these questions we need probability distributions 3

14 µ and σ of Distributions Define µ and σ in terms of the parent probability distribution P(x) Definition of P(x) Limit as The number of observations d that yield values between x and x + dx is d/ = P(x) dx 4

15 Expectation Values The mean, µ, is the expectation value <x> The variance, σ, is the expectation value <(x-µ) > 5

16 Expectation Values For a discrete distribution,, observations and n distinct outcomes µ = Lim = Lim = Lim x i i= n [ x j P(x j )] j = n [ x j P(x j )] j = 6

17 Expectation Values For a discrete distribution,, observations and n distinct outcomes σ = Lim = Lim = Lim i= n j = (x i µ) (x i µ) P(x i ) n [ (x j µ) P(x j )] j = 7

18 Expectation values The expectation value of any continuous function of x f (x) = µ = xp(x)dx σ = f (x)p(x)dx (x µ) P(x)dx where P(x)dx = 8

19 Binomial Distribution Suppose we have two possible outcomes with probability p and q = -p e.g., a coin toss, p = /, q = / / If we flip n coins what is the probability of getting x heads? h t Answer is given by the Binomial Distribution P(x;n, p) = C(n, x) p x q n x C(n, x) is the number of combinations of n items taken x at a time = n!/[x!(n-x)!] 9

20 Binomial Distribution The expectation value n µ = xp(x;n, p) x= 0 n = x C(n, x) p x q n x x= 0 n = x x= 0 n! x!(n x)! px ( p) n x = np 0

21 Poisson Distribution The Poisson distribution is the limit of the Binomial distribution when µ << n because p is small The binomial distribution describes the probability P(x; n, p) of observing x events per unit time out of n possible events Usually we don t know n or p but we do know µ

22 Poisson Distribution Suppose p << then x << n P(x;n, p) = n! x!(n x)! n! x!(n x)! px ( p) n x = n(n )(n )...(n x )(n x ) n x n! (n x)! px (np) x = µ x when n >> x ( p) n x = ( p) x ( p) n ( p) n since p << Lim ( p 0 p)n = Lim p 0 p [( p)/ ] µ = e ( ) µ = e µ P(x,µ) = µx x! e µ

23 Poisson Distribution The expectation value of x is x = xp(x,µ) = x x= 0 µx x= 0 x! e µ = µ Expectation value of (x-µ) σ = x µ ( ) = (x µ) x= 0 µ x x! e µ = µ 3

24 Gaussian or ormal Distribution The Gaussian distribution is an approximation to the binomial distribution for large n and large np P(x;µ,σ) = σ e π x µ σ 4

25 Gaussian or ormal Distribution P(x;µ,σ) = σ x µ π e σ P(x;µ,σ) = σ x µ π e σ π e x dx = /- σ: 68.3% +/- σ: 95.5% +/- 3σ: 99.7% 5

26 Combining Two Observations Suppose I have two sets of measurements, a i, and b i A derived quantity c i = a i + b i What is the relation between the means and standard deviations of a i and b i and c i Suppose we have the same number of observations 6

27 Combining Two Observations = a = b a = a i b = b i c = c i s c = ( c i c ) c i = a i + b i c = (a i + b i ) = a i + b i = a + b 7

28 Combining Two Observations s c = s c = = = ( c i c ), c = a + b [ a i + b i ( a + b )] [ ( a i + b i ) ( a i + b i )( a + b ) + ( a + b ) ] [ a i + b i + a i b i ( a i a + a i b + b i a + b i b ) + (a) + a b + ( b ) ] = a + b + a i b i (a ) a b ( b ) 8

29 Combining Two Observations s c = = a + = ( c i c ), c = a + b b + [ ] a i b i a (a) b b s a s c = s a + s b + s ab [ ( ) ] s b (a ) a b ( ab a b ) s ab ( b ) The term s ab is the covariance Murphy s law factor s ab can be negative, zero or positive 9

30 Combining Two Uncorrelated Observations When a and b are uncorrelated the covariance is zero s ab = ( a i a ) b i b = 0 ( ) s c = s a + s b The variance of c is the sum of the variances of a and b This demonstrates the fundamentals of error propagation 30

31 Propagation of Errors Suppose we want to determine x which is a function of measured quantities, u, v, etc. Assume that x = f (u,v,...) x = f (u,v,...) 3

32 Propagation of Errors The uncertainty in x can be found by considering the spread of the values of x resulting from individual measurements, u i, v i, etc., x i = f (u i,v i,...) In the limit of the variance of x σ x = Lim i ( x x ) i 3

33 Propagation of Errors Taylor expand the deviation x i x = σ x = Lim ( u i u ) f + u u = Lim i i ( v i v ) f +... v v ( u i u ) f + u u ( u i u ) f u u ( v i v ) f + v i v +... v v v ( ) f v + ( u i u )( v i v ) f f u u... v v 33

34 Propagation of Errors σ x = Lim = Lim Lim Lim i i i σ f x = σ u u ( u i u ) f u ( u i u ) f u ( v i v ) f v v u u + + ( ) f + v i v ( u i u )( v i v ) f f u u v +... v i u f + σ v v v v + σ uv f u u f v v +... v + ( u i u )( v i v ) f f u u... v v 34

35 Examples of Error Propagation Suppose a = b + c We know that a = b + c σ a = σ b + σ c assuming that the covariance is 0 What about a = b/c? 35

36 Examples of Error Propagation Suppose a = b/c? a = b c and σ a a = σ b b σ a = σ b c + σ c or σ a b a = σ b b + σ c c f + σ c v b c c a f + σ bc +... b b c c assuming that the covariance is 0 36

37 Examples of Error Propagation What happens when m = -.5 log 0 (F/F 0 )? What is the error in m? m =.5log 0 ( F F 0 ) and σ m m = σ F F σ.5 m = σ F F log 0 σ m =.087 F ( ) ( ) σ F F 37

38 Error of the Mean Suppose we have measurements, x i with uncertainties characterized by s i x = ( x + x + x x ) 3 = x i i s x x = s x x = s x i x i i x + s x x x x + s 3 x 3 x x s x x 38

39 Error of the Mean Suppose the errors on all points are equal so that s i = s s x s x = s x i x i i x x i = = s i x x i j x j = = s 39