ES 07 These slides can be found at optimized for Windows)

ES 07 The Gaussian distribution f(x)  dx is the probability that an observation will fall in between x – dx/2 and x + dx/2

ES 07 Normally the Gaussian distribution is standardized by putting  = 0 and  = 1  (  ) is called the frequency function note that  (  ) =  (–  )

ES 07 The distribution function is the primitive function to the frequency function The distribution function cannot be calculated analytically, but is tabulated in most standard books. Or it can be approximated. Note that  (–  ) = 1 –  (  ) and that F(x) =  (  ) The probability to obtain a value between  1 and  2 (  1 <  2 ) is given by  (  2 ) –  (  1 )

ES 07 An approximation which can be used for  (  ) is:  (  ) = 1 –  (  ) (a 1 t + a 2 t 2 +a 3 t 3 ) +  (  ) (   0) where t = (1 + p  ) -1 with p = a 1 = a 2 = a 3 = giving |  (  )| < 1  10 -5

ES 07 Expectation value, variance and covariance The sum is over the whole population Standard deviation:

ES 07 Variance of the population mean value V[  X  ] = V[X]/N

ES 07 Expectation value and variance from a sample Estimates with correct expectation value are thus given by: and

ES 07 The variance of the variance

ES 07 This leads to an estimate of the “error” in the estimate of the standard deviation of a distribution Beware! V[V[  X  ]] is normally a small positive number, but the terms used for its calculation are normally very large. High precision is needed in the calculations.

ES 07 Parameter fitting with the maximum likelihood method If we know that the sample we want to study comes from a certain distribution, e. g. a Gaussian with unknown parameters, we can fit those using he maximum likelihood method. Calculate the probability to obtain exactly the sample you have as a function of the parameters and maximize this probability L ( ,  ) =  f(x i ) or l ( ,  ) =  ln(f(x i )) The “error”  of a parameter p is estimated by l (p   p ) = l max – ½

ES 07 The l -function is usually close to a parabola l (p   p ) = l max – ½

ES 07  2 fitting and  2 testing This method needs binning of the data. In each bin we have (x i ) min, (x i ) max, y i = n i /N and  i which can be taken as  (n i )/N as long as n i  5 (no less than five observations in a bin) and n i  N. Minimize the sum S

ES 07 y th is calculated from the tested distribution. If this is a Gaussian with parameters  G and  G we have

ES 07 S can now be minimized with respect to the parameters to be fitted. When S min is found the “error” of the parameter can be estimated from (c. f. maximum likelihood method) S(p   p ) = S min + 1 S is in many cases approximately of parabolic shape close to the minimum.

ES 07 S is  2 distributed with degrees of freedom. The number of degrees of freedom are the number of bins we have minus the number of parameters that are fitted. In the previous example we had 7 bins and two parameters giving = 5. S( =5) Tablemeaning in only about 10 % of the cases a smaller S-value would be obtained in about 33 % of the cases a smaller S-value would be obtained in about 42 % of the cases a larger S-value would be obtained in about 17 % of the cases a larger S-value would be obtained in only about 8 % of the cases a larger S-value would be obtained Generally we expect S/ to be close to 1 if the fluctuations in the data are of purely statistical origin and if the data is described by the distribution in question.

ES 07 Confidence levels and confidence intervals Assume that we have estimated a parameter p and found that p = 1.23 with  p = Lets say that we want to construct an interval that covers the true value of p with 90 % confidence. This means that we leave out 5 % on each side. Start by finding , so that  (  ) = 0.95   =  p max =  = 1.41 and p min =  = 1.05 We have found the two sided confidence interval of our estimate of p on the 90 % confidence level to be 1.05 – 1.41

ES 07 If we want to state that p < x with some confidence we can construct a one sided confidence region. Lets say that we want to construct an region that covers the true value of p with 99 % confidence. Start by finding , so that  (  ) = 0.99   =  p max =  = 1.49 We have found the one sided confidence region of our estimate of p on the 99 % confidence level to be p < 1.49

ES 07 Hypothesis testing (simple case) Lets again assume that we have estimated a parameter p and found that p = 1.23 with  p = No we have a hypothesis stating that p = 1.4 We now ask our selves with what probability the hypothesis is wrong. We calculate  = (1.4 – 1.23)/0.11 = and the probability is given by  (  ) = 0.939, i. e. we can state with 94 % confidence that the hypothesis is wrong.

ES 07