Appendix A Numerical Tables

Transcription

1 Appendix A Numerical Tables A.1 The Gaussian Distribution The Gaussian distribution (Eq. 2.31)isdefinedas 1 f (x)= 2πσ 2 e (x μ) 2 2σ 2. (A.1) The maximum value is obtained at x = μ, and the value of (x μ)/σ at which the Gaussian is a times the peak value is given by z x μ σ = 2lna. (A.2) Figure A.1 shows a standard Gaussian normalized to its peak value, and values of a times the peak value are tabulated in Table A.1. The Half Width at Half Maximum (HWHM) has a value of approximately 1.18σ. A.2 The Error Function and Integrals of the Gaussian Distribution The error function is defined in Eq as erf z = 1 z π z e x2 dx (A.3) and it is related to the integral of the Gaussian distribution defined in Eq. 2.32, μ+zσ A(z)= μ zσ f (x)dx = 1 z 2π z e x2 /2 dx. (A.4) M. Bonamente, Statistics and Analysis of Scientific Data, Graduate Texts in Physics, DOI / , Springer Science+Business Media New York

2 222 A Numerical Tables Fig. A.1 Normalized values of the probability distribution function of a standard Gaussian (μ = 0andσ = 1) Table A.1 Values of a times the peak value for a Gaussian distribution a z a z a z a z a z The relationship between the two integrals is given by erf ( 2z)=A(z). (A.5) The function A(z) describes the integrated probability of a Gaussian distribution to have values between μ zσ and μ + zσ. The number z therefore represents the number of σ by which the interval extends in each direction. The function A(z) is tabulated in Table A.2, where each number in the table corresponds to a number z given by the number in the left column (e.g., 0.0, 0.1, etc.), and for which the second decimal digit is given by the number in the top column (e.g., the value of corresponds to z = 0.01). The cumulative distribution of a standard Gaussian function was defined in Eq as z 1 B(z)= e t2 2 dt; 2π (A.6)

3 A.2 The Error Function and Integrals of the Gaussian Distribution 223 Table A.2 Values of the integral A(z) as a function of z, the number of standard errors σ (continued)

4 224 A Numerical Tables Table A.2 (continued) a Table foot note (with superscript)

5 A.4 The χ 2 Distribution 225 and it is therefore related to the integral A(z) by B(z)= A(z) 2. (A.7) The values of B(z) are tabulated in Table A.3. Each number in the table corresponds to a number z given by the number in the left column (e.g., 0.0, 0.1, etc.), and for which the second decimal digit is given by the number in the top column (e.g., the value of corresponds to z = 0.01). Critical values of the standard Gaussian distribution functions corresponding to selected values of the integrals A(z) and B(z) are shown in Table A.4. They indicate the value of the variable z required to include a given probability, and are useful for either two-sided or one-sided rejection regions in hypothesis testing. A.3 Upper and Lower Limits for a Poisson Distribution The Gehrels approximation described in [15] can be used to calculate upper and lower limits for a Poisson distribution, when n obs counts are recorded. The confidence level is described by the parameter S, corresponding to the number of standard deviations σ for a Gaussian distribution; for example, S = 1 corresponds to an 84.1% confidence level, S = 2 to a 97.7%, and S = 3 corresponds to 99.9%; see Table 4.3 for correspondence between values of S and probability. The upper and lower limits are described, in the simplest approximation, by λ up = n obs + S S n 4 obs + 3 ( 4 λ lo = n obs 1 1 ) S 3 (A.8) 9n obs 3 n obs and more accurate approximations are provided in [15](TablesA.5 and A.6). A.4 The χ 2 Distribution The probability distribution function for a χ 2 variable is defined in Eq as ( ) 1 f /2 f χ 2 (z)= 1 2 Γ ( f /2) e z/2 z f /2 1, where f is the number of degrees of freedom. The critical value or p-quantile of the distribution is given by P χ 2 (z χ2 crit )= χ 2 crit 0 f χ 2 (z)dz = p (A.9)

6 226 A Numerical Tables Table A.3 Values of the integral B(z) as a function of z (continued)

7 A.4 The χ 2 Distribution 227 Table A.3 (continued) a Table foot note (with superscript)

8 228 A Numerical Tables Table A.4 Table of critical values of the standard Gaussian distribution to include a given probability, for two-sided confidence intervals ( z,z)ofthe integral A(z), andforor one-sided intervals (,z)of the integral B(z) Probability Two-sided z One-sided z Table A.5 Selected upper limits for a Poisson variable using the Gehrels approximation Upper limits Poisson parameter S or confidence level S = 1 S = 2 S = 3 n obs (1-σ, or 84.1%) (2-σ, or 97.7%) (3-σ, or 99.9%)

9 A.4 The χ 2 Distribution 229 Table A.6 Selected lower limits for a Poisson variable using the Gehrels approximation Lower limits Poisson parameter S or confidence level S = 1 S = 2 S = 3 n obs (1-σ, or 84.1%) (2-σ, or 97.7%) (3-σ, or 99.9%) or, equivalently, P χ 2 (z χ2 crit )= χ 2 crit f χ 2 (z)dz = 1 p. (A.10) The critical value is a function of the number of degrees of freedom f and the level of probability p. Normally p is intended as a large number, such as 0.68, 0.90 or 0.99, meaning that the value χcrit 2 leaves out just a 32, 10 or 1% probability that the value can be even higher. When using the χ 2 distribution as a fit statistic, the p quantile, or critical value with probability p, determines χ 2 χcrit 2 as the rejection region with probability p. For example, for f = 1 degree of freedom, χ is the 90% rejection region, and a best-fit value of the statistic larger than 2.7 would force a rejection of the model at the 90% level. As described in Sect. 5.2, theχ 2 distribution has the following mean and variance: { μ = f σ 2 = 2 f.

10 230 A Numerical Tables It is convenient to tabulate the value of reduced χ 2,orχcrit 2 / f, that corresponds to a give probability level, as function of the number of degrees of freedom. Selected critical values of the χ 2 distribution are reported in Table A.7. If Z is a χ 2 -distributed variable with f degrees of freedoms, Z f lim = N(0,1). f 2 f (A.11) In fact, a χ 2 variable is obtained as the sum independent distributions (Sect. 5.2), to which the central theorem limit applies (Sect. 3.3). For a large number of degrees of freedom, the standard Gaussian distribution can be used to supplement Table A.7 according to Eq. A.11. For example, for p = 0.99, the one-sided critical value of the standard Gaussian is approximately 2.326, according to Table A.4. Using this value into Eq. A.11 for f = 200 would give a critical value for the χ 2 distribution of (compare to from Table A.7). A.5 The F Distribution The F distribution with f 1, f 2 degrees of freedom is defined in Eq as f F (z)= Γ ( f 1 + f 2 ) 2 Γ ( f 1 2 )Γ ( f 2 2 ) ( f1 f 2 ) f f z 2 1 ( 1 + z f ) f 1 + f f 2. The critical value F crit that includes a probability p is given by P(z F crit )= f F (z)dz = 1 p, F crit (A.12) and it is a function of the degrees of freedom f 1 and f 2.InTableA.8 are reported the critical values for various probability levels p, for a fixed value f 1 = 1, and as function of f 2.TablesA.9 through A.15 have the critical values as function of both f 1 and f 2. Asymptotic values when f 1 and f 2 approach infinity can be found using Eq. 5.23: lim f 2 f F (z, f 1, f 2 )= f χ 2 (x, f 1 ) where x = f 1 z lim f 1 f F (z, f 1, f 2 )= f χ 2 (x, f 2 ) where x = f 2 /z. For example, the critical values of the F distribution for f 1 = 1 and in the limit of large f 2 are obtained from the first row of Table A.7.

11 A.5 The F Distribution 231 Table A.7 Critical values of the χ 2 distribution Probability p to have a value of reduced χ 2 below the critical value f

12 232 A Numerical Tables Table A.8 Critical values of F statistics for f 1 = 1 degrees of freedom Probability p to have a value of F below the critical value f , A.6 The Student s t Distribution The Student t distribution is given by Eq. 5.27, ( f T (t)= 1 Γ (( f + 1)/2) 1 + t2 f π Γ ( f /2) f ) 1 2 ( f +1), where f is the number of degrees of freedom. The probability p that the absolute value of a t variable exceeds a critical value T crit is given by P( t T crit )=P( x μ T crit s/ Tcrit n)= f T (t)dt = 1 p. T crit (A.13) These two-sided critical values are tabulated in Tables A.16 A.22 for selected values of f, as function of the critical value T. In these tables, the left column indicates the value of T to the first decimal digit, and the values on the top column are the second decimal digit. Table A.23 provides a comparison of the probability p for 5 critical values, T = 1 through 5, as function of f. The case of f = corresponds to a standard Gaussian.

13 A.6 The Student s t Distribution 233 Table A.9 Critical values of F statistic that include p = 0.50 probability f 1 f Table A.10 Critical values of F statistic that include p = 0.60 probability f 1 f

14 234 A Numerical Tables Table A.11 Critical values of F statistic that include p = 0.70 probability f 1 f Table A.12 Critical values of F statistic that include p = 0.80 probability f 1 f

15 A.6 The Student s t Distribution 235 Table A.13 Critical values of F statistic that include p = 0.90 probability f 1 f Table A.14 Critical values of F statistic that include p = 0.95 probability f 1 f

16 236 A Numerical Tables Table A.15 Critical values of F statistic that include p = 0.99 probability f1 f , , , , , , , , , ,

17 A.6 The Student s t Distribution 237 Table A.16 Integral of Student s function (f = 1), or probability p, as function of critical value Tcrit (continued)

18 238 A Numerical Tables Table A.16 (continued)

19 A.6 The Student s t Distribution 239 Table A.17 Integral of Student s function (f = 2), or probability p, as function of critical value Tcrit (continued)

20 240 A Numerical Tables Table A.17 (continued)

21 A.6 The Student s t Distribution 241 Table A.18 Integral of Student s function (f = 3), or probability p, as function of critical value Tcrit (continued)

22 242 A Numerical Tables Table A.18 (continued)

23 A.6 The Student s t Distribution 243 Table A.19 Integral of Student s function (f = 5), or probability p, as function of critical value Tcrit (continued)