# A survey of tables of probability distributions.

Key words: continuous univariate distributions; discrete univariate distributions; multivariate distributions; probability distributions.

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1. Introduction

Probabilities and percentiles of statistical probability distributions have historically been cited from reference tables published in books, journals, and other publications. Reference tables of probability distributions continued to be published from the 1920s through the 1980s and early 1990s. Some tables superceded their earlier counterparts. Abramowitz and Stegun  surveyed the tables published before 1964, and reproduced some of them. In particular, Abramowitz and Stegun  reproduced the tables of percentiles of chi-square, t-, and F-distributions from the 1954 edition of Pearson and Hartley . Other collections of tables of probability distributions include Greenwood and Hartley  and Owen .

This article is a survey of the tables published about or after 1964. A few earlier tables are also mentioned when appropriate. Most of the tables abstracted in this article are referenced in Pearson and Hartley , Pearson and Hartley , Johnson, Kotz, and Kemp , Johnson, Kotz, and Balakrishnan , Johnson, Kotz, and Balakrishnan , Johnson, Kotz, and Balakrishnan , and Kotz, Balakrishnan, and Johnson . The abstracts presented here have been verified from the original sources, and in some cases corrections and additions were made. The next three sections contain the abstracts for discrete univariate, continuous univariate, and multivariate probability distributions.

A random variable is denoted by X, and x denotes a particular value of X. The cumulative distribution function of X is F(x) = Pr{X [less than or equal to] x}. The survival function of X is F(x) = 1-F(x) = Pr{X > x}. For a discrete random variable f(x) interpreted as Pr{X = x} is the probability mass function (pmf). For a continuous random variable f(x) interpreted as dF(x)/dx is the probability density function (pdf). A particular value, x, is the rth quantile of X when F(x) = r, for 0 [less than or equal to] r [less than or equal to] 1. The rth quantile is commonly referred to as the r X 100th percentile of X. The expected value or mean, and the variance of X are denoted by E(X), and V(X), respectively. The abbreviation nD, for an integer n, denotes n decimal places. An expression such as 0.01(0.02)0.09 denotes the sequence of numbers from 0.01 to 0.09 increasing in steps of 0.02. Log denotes natural logs unless indicated otherwise.

2. Discrete Univariate Distributions

2.1 Binomial Distribution

The pmf is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x = 0, 1,..., n.

Weintraub  tabulated, to 10D, [bar.F](x - 1) for p = 0.0001(0.0001) 0.0009, 0.001(0.001)0.1, and n = 1(1)100.

Pearson and Hartley  tabulated, to 5D, f(x) for p = 0.01, 0.02(0.02)0.10, 0.10(0.10)0.50, and n = 5(5)30.

2.2 Poisson Distribution

The pmf is

f(x) = [[e.sup.-[theta]][[theta].sup.x]]/x!

for x = 0, 1,....

Defense Systems Department, General Electric Company  tabulated, to 8D, f(x), F(x), and [bar.F](x - 1) for [theta] ranging from [10.sup.-7] to 205 with increments ranging from [10.sup.-7] to 5.

Khamis and Rudert  tabulated, to 10D, [bar.F](x - 1) for [theta] = 0.00005(0.00005)0.0005, 0.0005(0.0005)0.005, 0.005(0.005)0.5, 0.5(0.025)3, 3(0.05)8, 8(0.25)33, 33(0.5)83, and 83(1)125.

Pearson and Hartley  tabulated, to 6D, f(x) for [theta] = 0.1(0.1)15.0. They also tabulated, to 5D, F(x) for [theta] = 0.0005(0.0005)0.005, 0.005(0.005)0.05, 0.05(0.05)1, 1(0.1)5, 5(0.25)10, 10(0.5)20, 20(1)60, and x = 1(1)35.

2.3 Negative Binomial Distribution

The pmf is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x = 0, 1,... and E(X) = k(1 - p)/p.

Grimm  tabulated, to 5D, f(x) and F(x) for E(X) = 0.1(0.1)1.0, 1.0(0.2)4.0, 4.0(0.5)10.0, and 1/p = 1.2, 1.5, 2.0(1)5.

Williamson and Bretherton  tabulated, to 6D, f(x) and F(x) for the following values of p and k: p = 0.05 and k = 0.1(0.1)0.5, p = 0.10 and k = 0.1(0.1)1.0, p = 0.12(0.02)0.20 and k = 0.1(0.1)2.5, p = 0.22(0.02)0.40 and k = 0.1(0.1)2.5(0.5)5.0, p = 0.42(0.02)0.60 and k = 0.1(0.1)2.5(0.5)10.0, p = 0.62(0.02)0.80 and k = 0.2(0.2)5.0(1)20, p = 0.82(0.02)0.90 and k = 0.5(0.5)10.0(2)50, p = 0.95 and k = 2(2)50(10)200. Deahl  extended the Williamson and Bretherton  table of F(x) for p = 0.02, 0.04, 0.05, 0.06, 0.08, 0.10, and k = 0.10(0.10)2.00.

Brown  tabulated, to 4D, f(x) and F(x) for E(X) = 0.25(0.25)1.0, 1.0(1)10, and 1/p = 1.5(0.5)5.0, 5(1)7.

2.4 Hypergeometric Distribution

The pmf is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for max[0, n - N + k] [less than or equal to] x [less than or equal to] min[k, n].

Lieberman and Owen  tabulated, to 6D, f(x) and F(x) for N = 2(1)100, n = 1(1)50, and all possible values of k; N = 1000, n = 500, and all possible values of k; and N = 100(100)2000, n = N/2, and k = (n - 1).

2.5 Logarithmic Series Distribution

The pmf is

f(x) = [-[[theta].sup.x]]/[[log(1 - [theta])]x]

for x = 1, 2,... and E(X) = -[theta]/[(1 - [theta]) log(1 - [theta])].

Patil  tabulated, to 4D, E(X) as a function of [theta] for [theta] = 0.01(0.01)0.99. Patil, Kamat, and Wani  tabulated, to 6D, f(x) and F(x) for [theta] = 0.01(0.01)0.70, 0.70(0.005)0.900, and 0.900(0.001)0.999. Patil and Wani  tabulated, to 4D, parameter [theta] for E(X) = 1.02(0.02)2.00, 2.00(0.05)4.00, and 4.00(0.1)8.0, 8.0(0.2)16.0, 16.0(0.5)30.0, 30.0(2)40, 40(5)60, 60(10)140, and 140(20)200.

Williamson and Bretherton  tabulated, to 5D, f(x) and F(x) for E(X) = 1.1(0.1)2.0, 2.0(0.5)5.0, 5.0(1)10.0. They also tabulated [theta], to 5D, for E(X) = 1.0(0.1)10.0, and 10.0(1)50.

2.6 Neyman Type A Distribution

This is the Poisson-stopped-summed-Poisson distribution (Johnson, Kotz, and Kemp ). The pmf is

f(x) = [[[e.sup.-[lambda]][[phi].sup.x]]/x!] [[infinity].summation over (j=0)] [[([lambda][e.sup.-[phi]])[.sup.j] [j.sup.x]]/j!]

for x = 0, 1,... and E(X) = [lambda][phi].

Grimm  tabulated, to 5D, f(x) for E(X) = 0.1(0.1)1.0, 1.0(0.2)4, 6, 10, and [phi] = 0.2, 0.5, 1.0, 2, 3, 4 up to f(x) = 0.99900.

2.7 Geometric-Poisson Distribution

The pmf is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x = 1, 2,...; f(x) = [e.sup.-[theta]] for x = 0; and E(X) = [theta]/(1-p). This distribution is also called Polya-Aeppli distribution.

Sherbrooke  tabulated, to 4D, f(x) and F(x) for E(X) = 0.10, 0.25(0.25)1.00, 1.00(0.5)3.0, 3.0(1)10, and (1 + p)/(1 - p) = 1.5(0.5)5.0, 5.0(1)7.

3. Continuous Univariate Distributions

3.1 Standard Normal (Gaussian) Distribution

The pdf is

f(x) = [1/[square root of (2[pi])]] [e.sup.-[x.sup.2]/2].

Abramowitz and Stegun  tabulated the following: F(x), to 15D, for x = 0.00(0.02)3.00 and, to 10D, for x = 3.00(0.05)5.00; f(x), to 5D, for x such that [bar.F](x) = q for q = 0.000(0.001)0.500; and x such that [bar.F](x) = q for q = 0.000(0.001)0.500, 0.0000(0.0001)0.0250, and q = [10.sup.-m] for m = 4(1)23. They also tabulated the derivatives of f(x) up to the order 12.

Pearson and Hartley  tabulated, to 10D, quantiles x and corresponding f(x), where F(x) = p for p = 0.500(0.001)0.999, and 0.9990(0.0001)0.9999.

White  tabulated, to 20D, quantiles x such that [bar.F](x) = q for q = 0.005(0.005)0.500, and q = 5 X [10.sup.-k], 2.5 X [10.sup.-k], and 1 X [10.sup.-k], where k = 1(1)20.

3.2 Standardized Stable Distributions

The pdfs of standardized stable distributions are unimodal with shape depending on the parameters [beta] and [alpha]. Although the pdfs are rather complicated, they can be expressed as convergent series (Johnson, Kotz, and Balakrishnan ).

Fama and Roll  tabulated, to 4D, F(x) for [beta] = 0 and [alpha] = 1.0(0.1)1.9, 1.95, 2.0, and x = 0.05(0.05)1.00, 1.00(0.1)2.0, 2.0(0.2)4.0, 4.0(0.4)6.0, 6.0(1)8, 10, 15, and 20. They also tabulated, to 3D, quantiles x such that F(x) = p, for p = 0.52(0.02)0.94, 0.94(0.01)0.97, 0.97(0.005)0.995, and 0.9975.

Holt and Crow  tabulated, to 4D, f(x) for [beta] = -1.00(0.25)1.00 and [alpha] = 0.25(0.25)2.00, and nonnegative x in steps varying by factors of 10 from 0.001 to 100 such that interpolation is possible. The tabulation is terminated when f(x) first falls to 0.0001. The largest such value of x is 338, for [alpha] = 0.25 and [beta] = -1.00.

Worsdale  tabulated, to 4D, F(x) for [beta] = 0 and [alpha] = 0.6(0.1)2.0, and x = 0.00(0.05)3.00. For larger values of x, F(x) is tabulated for [log.sub.10] x = 0.40(0.05)2.50.

Panton  tabulated, to 5D, F(x) for [beta] = 0 and [alpha] = 1.0(0.1)2.0, and x = 0.05(0.05)1.00, 1.00(0.1)2.0, 2.0(0.2)4.00, 4.00(0.4)6.0, 7, 8, 10, 15, 20.

3.3 Inverse Gaussian Distribution

The pdf is

f(x) = ([[lambda]/[2[pi][x.sup.3]]])[.sup.1/2] exp([[-[lambda](x - [mu])[.sup.2]]/[2[[mu].sup.2]x]])

for x > 0, [lambda] > 0, [mu] > 0, and E(X) = [mu] and V(X) = [[mu].sup.3]/[lambda].

Wasan and Roy  tabulated, to 4D, quantiles x such that [mu] = t, [lambda] = [t.sup.2], that is, [mu] = t = V(X), and F(x) = p, where t = 0.1(0.1)4.0, 4.0(0.2)6.0, 6.0(1.0)35.0, 35(5)100, 100(10)150, 150(20)250, 300(100)1000, 1000(200)1600, 2000(400)4000, and p = 0.005, 0.010, 0.025(0.025)0.100, 0.25(0.25)0.75, 0.80, 0.900(0.025)0.975, and 0.990. In order to determine quantiles of an inverse Gaussian random variable Y with parameters [mu] > 0 and [lambda] > 0, use the fact that the distribution of X = [lambda] Y/[[mu].sup.2] is inverse Gaussian with parameters t and [t.sup.2], where t = [lambda]/[mu].

Chan, Cohen, and Whitten  tabulated F(x) of the standardized inverse Gaussian distribution with E(X) = 0 and V(X) = 1 for various values of the standardized third moment about the mean [[alpha].sub.3] = [square root of ([[beta].sub.1])]. They tabulated, to 6D, F(x) for x = -3.0(0.1)5.9 with [[alpha].sub.3] = 0.0(0.1)1.2, and x = -1.5(0.1)7.4 with [[alpha].sub.3] = 1.3(0.1)2.5.

Koziol  tabulated quantiles x, to eight significant digits, such that F(x) = p, [mu] = t, [lambda] = [t.sup.2], where t = 0.02(0.02)4, 4(0.04)6, 6(0.02)35, 35(1)100, 100(2)150, 150(4)250, 250(10)300, 300(20)600, 600(40)2000, 2000(80)4000, and p = 0.001, 0.005, 0.01(0.01)0.99, 0.995, 0.999.

3.4 Incomplete Gamma Function

Harter  tabulated, to 9D, the incomplete [GAMMA]-function ratio I(u, p) defined by Pearson  as

I(u, p) = [1/[[GAMMA](p + 1)]] [[integral].sub.0.sup.u[square root of (p+1)]] [[upsilon].sup.p][e.sup.-[upsilon]]d[upsilon]

for u at intervals of 0.1, starting from 0.0, and p = -0.5(0.5)74 and 74(1)164. Harter  extended Harter  for p = -0.95(0.05)4.

3.5 Standard Gamma Distribution

The pdf is

f(x) = [1/[[GAMMA]([alpha])]][x.sup.[alpha]-1][e.sup.-x]

for x [greater than or equal to] 0 and [alpha] > 0.

Wilk, Gnanadesikan, and Huyett  tabulated quantiles x, accurate to about five significant digits, for [alpha] = 0.1(0.1)0.6, 0.6(0.2)5.0, 5.0(0.5)10.0, 10.0(1.0)20.0, and p = 0.1, 0.5, 0.7, 1.0(0.5)3.0, 3.0(1.0)5.0, 7.5, 10.0(5.0)30.0, 30(10)70, 70(5)90, 90.0(2.5)97.5, 98.0, 99.0, 99.5, and 99.9.

Thom  tabulated, to 4D, F(x) for [alpha] = 0.5(0.5)15.0, 15(1)36, and x = 0.0001, 0.001, 0.004(0.002)0.020, 0.02(0.02)0.80, 0.8(0.1)2.0, 2.0(0.2)3.0, 3.0(0.5)9.0; also tabulated, to 4D, quantiles x and corresponding f(x) such that F(x) = p for [alpha] = 0.5(0.5)15.0, 15(1)36, and p = 0.01, 0.05(0.05)0.95, 0.99.

Harter  tabulated, to 5D, quantiles x such that F(x) = p against the coefficient of skewness [square root of ([[beta].sub.1])] = [[mu].sub.3]/[[sigma].sup.3] = 2/[square root of [alpha]] for [square root of [[beta].sub.1]] = 0.0(0.1)4.8, and 4.8(0.2)9.0, and p = 0.0001, 0.0005, 0.0010, 0.0050, 0.0100, 0.0200, 0.0250, 0.0400, 0.0500, 0.1000(0.1000)0.9000, 0.9500, 0.9600, 0.9750, 0.9800, 0.9900, 0.9950, 0.9990, 0.9995 and 0.9999. Harter  extended Harter  for p = 0.002000, 0.429624, 0.570376, and 0.998000.

3.6 Chi-Square Distribution

The pdf is

f(x) = [x.sup.(v-2)/2]/[[2.sup.v/2][GAMMA](v/2)[e.sup.-x/2]]

for x > 0 and degrees of freedom v > 0. If [X.sub.1], [X.sub.2],..., [X.sub.v] have independent standard normal distributions, then X = [[SIGMA].sub.i=1.sup.n][X.sub.i.sup.2] has a chi-square distribution with v degrees of freedom.

Harter  tabulated, to six significant digits, quantiles x such that F(x) = p for p = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05, 0.1(0.1)0.9, 0.95, 0.975, 0.99, 0.995, 0.999, 0.9995, and 0.9999, and v = 1(1)150, and 150(2)330. A subset of these tables for v = 1(1)100 is reproduced in Harter .

Khamis and Rudert  tabulated F(x), to 10D, for v = 0.1(0.1)20, 20(0.2)40, 40(0.5)140, and x = 0.0001(0.0001)0.001, 0.001(0.001)0.01, 0.01(0.01)1, 1(0.05)6, 6(0.1)16, 16(0.5)66, 66(1)166, 166(2)250.

Pearson and Hartley  tabulated, to six significant digits, quantiles x such that F(x) = p for p = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05, 0.1(0.1)0.9, 0.95, 0.975, 0.99, 0.995, 0.999, 0.9995, and 0.9999, and v = 0.1(0.1)3.0, 3.0(0.2)10.0, and 10(1)100.

Mardia and Zemroch  tabulated, to five significant digits, quantiles x such that F(x) = p for p = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.02, 0.025, 0.03(0.01)0.1, 0.2, 0.25, 0.3(0.1)0.7, 0.75. 0.8, 0.9(0.01).97, 0.975, 0.98, 0.99, 0.995, 0.999, 0.9995, 0.9999, and fractional degrees of freedom v = 0.1(0.1)3.0, 3.0(0.2)7.0, 7.0(0.5)11, 11(1)30, 30(5)60, 60(10)120.

3.7 Standardized Weibull Distribution

The pdf is

f(x) = [gamma][x.sup.[gamma]-1][e.sup.-x.sup.[gamma]]

for x > 0 and [gamma] > 0, where [gamma] is the shape parameter.

Plait  tabulated, to 8D, f(x) for [gamma] = 0.1(0.1)3, 3(1)10, and tabulated, to 7D, F(x) for [gamma] = 0.1(0.1)4.0.

3.8 Standardized Extreme Value Distribution--Type 1

The pdf is

f(x) = exp(-x-[e.sup.-x]).

Gumbel  tabulated, to 7D, f(x) and F(x) for the following values of x: -3.0(0.1)-2.4, -2.40(0.05)0.00, 0.0(0.1)4.0, 4.0(0.2)8.0, and 8.0(0.5)17.0. Also, tabulated, to 5D, quantiles x such that F(x) = p for p = 0.0001(0.0001)0.0050, 0.005(0.001)0.988, 0.9880(0.0001)0.9994, and 0.99940(0.00001)0.99999.

White  tabulated, to 7D, the means and variances of all order statistics for sample sizes 1(1)50 and 50(5)100. Extended tables of means, variances, and covariances of all order statistics for sample sizes up to 30 have been provided by Balakrishnan and Chan  and Balakrishnan and Chan .

3.9 Incomplete Beta Function

Pearson  tabulated, to 7D, the incomplete B-function ratio I(p, q) defined as

I(p,q) = [[[GAMMA](p + q)]/[[GAMMA](p)[GAMMA](q)]] [[integral].sub.0.sup.x] [t.sup.p-1] (1 - t)[.sup.q-1] dt

for p, q = 0.5(0.5)11.0(1)50 with p [greater than or equal to] q and x = 0.00(0.01)1.00. These values are reproduced in Pearson . Additional values of I(p,q) are given, to 7D, for p = 11.5(1.0)14.5, q = 0.5, and x = 0.00(0.01)1.00. More values of I(p, q) are given, to 7D, for p = 0.5(0.5)11.0(1)16, q = 1.0(0.5)3.0, and x = 0.975, 0.980, 0.985, 0.988(0.001)0.999. Even more values of I(p, q) are given for p = 0.5(0.5)11.0(1)16, q = 0.5, and x = 0.9750, 0.9800, 0.9850, 0.9880(0.0005)0.9985, 0.9988(0.0001)0.9999. For x [greater than or equal to] 0.988, values are given to 8D.

3.10 Beta Distribution

The pdf is

f(x) = [1/[B(a,b)]][x.sup.a-1](1 - x)[.sup.b-1]

for 0 < x < 1, a > 0, b > 0 and B(a, b) = [GAMMA](a)[GAMMA](b)/[GAMMA](a + b).

Harter  tabulated, to 7D, quantiles x such that F(x) = p for a = 1(1)40, b = 1(1)40, and p = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05, 0.1(0.1)0.9, 0.95, 0.975, 0.99, 0.995, 0.999, 0.9995, 0.9999.

Vogler  tabulated, to six significant digits, quantiles x such that F(x) = p for a = 0.50(0.05)1.00, 1.1, 1.25(0.25)2.50, 2.50(0.5)5.0, 6, 7.5, 10, 12, 15, 20, 30, 60, b = 0.5(0.5)5.0, 6, 7.5, 10, 12, 15, 20, 30, 60, and p = 0.0001, 0.001, 0.005, 0.01, 0.025, 0.05, 0.1, 0.25, 0.5.

Pearson and Hartley  tabulated, to five significant digits, quantiles x such that F(x) = p for a = 0.5(0.5)15.0, 20, 30, 60, b = 0.5(0.5)5.0, 6, 7.5, 10, 12, 15, 20, 30, 60, and p = 0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 0.10, 0.25, 0.50.

3.11 F-Distribution

If [X.sub.1] and [X.sub.2] have independent chi-square distributions with degrees of freedom [v.sub.1] and [v.sub.2], respectively, then

X = [[X.sub.1]/[v.sub.1]]/[[X.sub.2]/[v.sub.2]]

has an F-distribution with [v.sub.1] (numerator) and [v.sub.2] (denominator) degrees of freedom.

Pearson and Hartley  tabulated, to five significant digits, quantiles x such that F(x) = p for p = 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999, and [v.sub.1] = 1(1)10, 12, 15, 20, 24, 30, 40, 60, 120, [infinity], and [v.sub.2] = 1(1)30, 40, 60, 120, [infinity].

Mardia and Zemroch  tabulated, to five significant digits, quantiles x such that F(x) = p for p = 0.5, 0.6, 0.7, 0.75, 0.8, 0.90(0.01)0.99, 0.975, 0.995, 0.999, 0.9995, 0.9999, and [v.sub.1] = 0.1(0.1)1.0, 1.0(0.2)2.0, 2.0(0.5)5, 5(1)16, 18, 20, 24, 30, 40, 60, 120, [infinity], and [v.sub.2] = 0.1(0.1)3.0, 3.0(0.2)7.0, 7.0(0.5)11, 11(1)40, 60, 120, [infinity]. A part of this table is reproduced in Pearson and Hartley .

3.12 t-Distribution

If [X.sub.1] has the standard normal distribution and [X.sub.2] has an independent chi-square distribution with v degrees of freedom, then

X = [X.sub.1] / [square root of ([X.sub.2]/v)]

has a Student's t-distributon with v degrees of freedom.

Fisher and Yates  tabulated, to 3D, quantiles x such that F(x) = p for p = 0.55(0.05)0.95, 0.975, 0.99, 0.995, 0.9995, and v = 1(1)30, 40, 60, 120. Lempers and Louter  extend these tables for p = 0.5625(0.0625)0.9375.

Hill  tabulated, to 20D or 20 significant digits, quantiles x such that F(x) = p/2 where p = 0.9(-0.1)0.1, [10.sup.-m], 2 X [10.sup.-m], 5 X [10.sup.-m], for m = 2(1)10(5)30, and v = 1(1)30, 30(2)50, 50(5)100, 100(10)150, 200, [240, 300, 400, 600, 1200] X [1, 10, 100], and [infinity].

Mardia and Zemroch  tabulated, to five significant digits, quantiles x such that F(x) = p for p = 0.5, 0.6, 0.7, 0.75, 0.8, 0.90(0.01)0.99, 0.975, 0.995, 0.999, 0.9995, 0.9999, and fractional degrees of freedom v = 0.1(0.1)3.0, 3.0(0.2)7.0, 7.0(0.5)11, 11(1)40, 60, 120, and [infinity].

3.13 Noncentral Chi-Square Distribution

If [X.sub.1], [X.sub.2],..., [X.sub.v] have independent standard normal distributions and [[delta].sub.1], [[delta].sub.2],..., [[delta].sub.v] are constants then

X = [v.summation over (i=1)]([X.sub.1] + [[delta].sub.i])[.sup.2]

has a noncentral chi-square distribution with v degrees of freedom and noncentrality parameter [lambda] = [[SIGMA].sub.i=1.sup.v][[delta].sub.i.sup.2].

Johnson  tabulated, to four significant digits, quantiles x such that F(x) = p for p = 0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 0.10, 0.25, 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999, v = 1(1)12, 15, 20, and square root of the noncentrality parameter [square root of [lambda]] = 0.2(0.2)6.0.

Haynam, Govindarajulu, and Leone  tabulated the power 1 - [beta] of chi-square test of significance as a function of the level of significance [alpha], degrees of freedom v, and noncentrality parameter [lambda] for [alpha] = 0.001, 0.005, 0.01, 0.025, 0.05, 0.1, v = 1(1)30, 30(2)50, 50(5)100, and [lambda] = 0.0(0.1)1.0, 1.0(0.2)3.0, 3.0(0.5)5.0, 5(1)40, 40(2)50, 50(5)100. They also tabulated the noncentrality parameter [lambda] as a function of [alpha], v, and 1 - [beta] for the values of [alpha] and v listed above and 1 - [beta] = 0.1(0.02)0.7, 0.7(0.01)0.99.

Pearson and Hartley  tabulated, to 3D, noncentrality parameter [lambda] as a function of the level of significance [alpha], degrees of freedom v, and power 1 - [beta] for [alpha] = 0.05, 0.01, v = 1(1)30, 30(2)50, 50(5)100, and 1 - [beta] = 0.25, 0.50, 0.60, 0.70(0.05)0.95, 0.97, 0.99.

3.14 Noncentral Chi Distribution

If [X.sub.1] has a noncentral chi-square distribution then the distribution of X = [square root of [X.sub.1]] is referred to as noncentral chi distribution.

Johnson and Pearson  tabulated, to four significant digits, quantiles x of chi distribution such that F(x) = p for p = 0.005, 0.01, 0.025, 0.05, 0.95, 0.975, 0.99, 0.995, degrees of freedom v = 1(1)12, 15, 20, and square root of the noncentrality parameter [square root of [lambda]] = 0.0(0.2)6.0. Approximate quantiles to three significant digits are also given for [square root of [lambda]] = 8.0 and 10.0. These tables are reproduced in Pearson and Hartley .

3.15 Noncentral F-Distribution

If [X.sub.1] has a noncentral chi-square distribution with [v.sub.1] degrees of freedom and noncentrality parameter [lambda], [X.sub.2] has a chi-square distribution with [v.sub.2] degrees of freedom, and [X.sub.1] and [X.sub.2] are independently distributed then

X = [[X.sub.1]/[v.sub.1]]/[[X.sub.2]/[v.sub.2]]

has a noncentral F-distribution with [v.sub.1] and [v.sub.2] degrees of freedom and noncentrality parameter [lambda].

Tiku  tabulated, to 4D, the power of the F-test for the level of significance [alpha] = 0.005, 0.01, 0.025, 0.05, [v.sub.1] = 1(1)10, 12, and [v.sub.2] = 2(2)30, 40, 0, 120, [infinity], and noncentrality parameter [lambda] such that [square root of ([lambda]/([v.sub.1] + 1))] = 0.5, 1.0(0.2)2.2, 2.2(0.4)3.0.

3.16 Doubly Noncentral F-Distribution

If [X.sub.1] has a noncentral chi-square distribution with [v.sub.1] degrees of freedom and noncentrality parameter [[lambda].sub.1], [X.sub.2] has a noncentral chi-square distribution with [v.sub.2] degrees of freedom and noncentrality parameter [[lambda].sub.2], and [X.sub.1] and [X.sub.2] are independently distributed then

X = [[X.sub.1]/[v.sub.1]]/[[X.sub.2]/[v.sub.2]]

has a doubly noncentral F-distribution with [v.sub.1] and [v.sub.2] degrees of freedom, and noncentrality parameters [[lambda].sub.1] and [[lambda].sub.2].

Tiku  tabulated, to 4D, the power of the F-test for the level of significance [alpha] = 0.01 and 0.05, degrees of freedom [v.sub.1] = 1(1)8, 10, 12, 24 and [v.sub.2] = 2(2)12, 16, 20, 24, 30, 40, 60, noncentrality parameters [[lambda].sub.1] and [[lambda].sub.2] such that [[phi].sub.1] = [square root of ([[lambda].sub.1]/([v.sub.1] + 1))] = 0(0.5)3 and [[phi].sub.2] = [[lambda].sub.2]/[square root of [v.sub.2]] = 0(1)8. Tiku  also tabulated, to 4D, the power of F-test for the critical values [F.sub.0] such that [u.sub.0] = 1/[1 + ([v.sub.1]/[v.sub.2])[F.sub.0]] = 0.02(0.08)0.50, 0.60, 0.75, 0.95, degrees of freedom [v.sub.1] = [v.sub.2] = 4(2)12, and the same noncentrality parameters as used in the previous table.

3.17 Noncentral t-Distribution

If [X.sub.1] has the standard normal distribution and [X.sub.2] has an independent chi-square distribution with v degrees of freedom, then

X = ([X.sub.1] + [delta])/[square root of ([X.sub.2]/v)]

has a noncentral t-distributon with v degrees of freedom and noncentrality parameter [delta].

Bagui  tabulated, to 5D, quantiles x of noncentral t-distribution such that F(x) = p for p = 0.01, 0.025, 0.05, 0.10, 0.20, 0.30, 0.70, 0.80, 0.90, 0.95, 0.975, 0.99, degrees of freedom v = 1(1)60, and noncentrality parameter [delta] = 0.1(0.1)8.0.

3.18 Doubly Noncentral t-Distribution

If [X.sub.1] has the standard normal distribution and [X.sub.2] has an independent noncentral chi-square distribution with v degrees of freedom and noncentrality parameter [lambda], then

X = ([X.sub.1] + [delta])/[square root of ([X.sub.2]/v)]

has a doubly noncentral t-distributon with v degrees of freedom, numerator noncentrality parameter [delta], and denominator noncentrality parameter [lambda].

Bulgren  tabulated, to 6D, F(x) of doubly non-central t-distribution with degrees of freedom v = 2(1)20, absolute value of numerator noncentrality parameter |[delta]| = 0(1)5, denominator noncentrality parameter [lambda] = 0, 1, 2(2)8, and x = 0.0, 0.1, 0.2(0.2)9.0.

3.19 Distribution of the Sample Correlation Coefficient From Bivariate Normal Distribution

Suppose ([Y.sub.i], [Z.sub.i]), for i = 1, 2,..., n, are independently distributed and have a common joint bivariate normal distribution with correlation coefficient [rho]. Then the sample correlation coefficient

X = [[[SIGMA].sub.i=1.sup.n]([Y.sub.i] - [bar.Y])([Z.sub.i] - [bar.Z])]/[square root of ([[SIGMA].sub.i=1.sup.n]([Y.sub.i] - [bar.Y])[.sup.2][[SIGMA].sub.(i=1).sup.n]([Z.sub.i] - [bar.Z])[.sup.2])]

has a distribution that depends only on the correlation coefficient [rho] and the sample size n.

Odeh  tabulated, to 5D, quantiles x of the sample correlation coefficient, where F(x) = p for p = 0.005, 0.01, 0.025, 0.05, 0.10, 0.25, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, [rho] = 0.0(0.10)0.90, 0.95, and n = 4(1)30, 30(2)40, 40(5)50, 50(10)100, 100(20)200, and 200(100)1000.

3.20 Distribution of the Sample Multiple Correlation Coefficient From Multivariate Normal Distribution

If the random variables [X.sub.1],..., [X.sub.M] have a joint multivariate normal distribution, then the smallest mean squared error linear predictor of [X.sub.1] is the conditional expected value E([X.sub.1]|[x.sub.2],..., [x.sub.M]). The multiple correlation coefficient R is the correlation between [X.sub.1] and its smallest mean squared error linear predictor. The distribution of the sample multiple correlation coefficient r depends only on the population coefficient R, number of variates M, and the sample size N.

Pearson and Hartley  tabulated, to 3D, lower and upper 1 and 5 percent points of the sample multiple correlation coefficient for R = 0.1(0.1)0.9, the sample size N such that N - M = 10(10)50, and M - 1 = 2(2)12, 12(4)24, 30, 34, 40.

4. Multivariate Distributions

4.1 Multivariate Normal Distribution

The multivariate normal density function of the random vector ([X.sub.1],..., [X.sub.M]) is

f([x.sub.1],..., [x.sub.M]) = [[|[SIGMA]|[.sup.-1/2]]/[(2[pi])[.sup.M/2]]]exp[-(1/2)(x - [mu])'[[SIGMA].sup.-1](x - [mu])],

where x = ([x.sub.1],..., [x.sub.M])', [mu] = ([[mu].sub.1],..., [[mu].sub.M])' is the mean vector, and [SIGMA] = [[[sigma].sub.ij]] is the positive definite covariance matrix. Here ([x.sub.1],..., [x.sub.M])', denotes transpose of the vector ([x.sub.1],..., [x.sub.M]). For the case [mu] = (0,..., 0)', [[sigma].sub.ii] = 1, [[sigma].sub.ij] = [rho], where 0 [less than or equal to] p < 1 and i, j = 1,..., M, i [not equal to] j, a one-sided upper equicoordinate p X 100 percentage point g is such that

Pr{[max.[1[less than or equal to]i[less than or equal to]M]] [X.sub.i] [less than or equal to] g} = p,

and a two-sided upper equicoordinate p X 100 percentage point h is such that

Pr{[max.[1[less than or equal to]i[less than or equal to]M]] |[X.sub.i]| [less than or equal to] h} = p,

where p is a specified value for the probability integral.

Gupta  tabulated equicoordinate one-sided probabilities p, to 5D, for g = -3.5(0.1)3.5, M = 1(1)12, and [rho] = 0.100, 0.125, 0.200(0.05)0.300, 1/3, 0.375, 0.400(0.1)0.600, 0.625, 2/3, 0.700(0.05)0.800, 0.875, and 0.900.

Tong  tabulated equicoordinate one-sided and two-sided percentage points, to 4D, and probability integrals p, to 5D. The table of one-sided percentage points gives the values of g for M = 2(1)20, [rho] = 0.0(0.1)0.9, 1/3, 2/3, 1/4, and 3/4, and p = 0.90, 0.95, and 0.99. The table of one-sided probability integrals gives the values of p for g = -2.0(0.1)4.0, M = 2(1)10, 10(2)20, and [rho] = 0.0(0.1)0.9, 1/3, 2/3, 1/4, and 3/4. The table of two-sided percentage points gives the values of h for the same set of M, [rho], and p as the one-sided percentage points. The table of two-sided probability integrals gives the values of p for h = 0.1(0.1)5.0 and the same set of M, and [rho] as the one-sided probability integrals.

4.2 Multivariate t-Distribution

Suppose the random vector Z = ([Z.sub.1],..., [Z.sub.M])' has a multivariate normal distribution with mean [mu] = 0 and covariance matrix [SIGMA] = [[[sigma].sub.ij]], where [[sigma].sub.ii] = 1 for i, j = 1,..., M (that is, [SIGMA] is a correlation matrix). Suppose S is a random variable independent of Z such that vS has a chi-square distribution with v degrees of freedom. Then the joint distribution of ([T.sub.1],..., [T.sub.M])' = ([Z.sub.1]/S,..., [Z.sub.M]/S)' is called a multivariate t-distribution with parameters [SIGMA] and v. A one-sided upper equicoordinate p X 100 percentage point g is such that

Pr{[max.[1[less than or equal to]i[less than or equal to]M]] [T.sub.i] [less than or equal to] g} = p,

and a two-sided upper equicoordinate p X 100 percentage point h is such that

Pr{[max.[1[less than or equal to]i[less than or equal to]M]] |[T.sub.i]| [less than or equal to] h} = p.

Freeman, Kuzmack, and Maurice  tabulated percentage points g to, 3D, for M = 2, and to, 2D, for M = 3, 4, 5, for p = 0.95, v = (M + 1)k for k = 9(10)99, 199, 499, and the following correlation structure: [[rho].sub.ij] = -1/2 for |i - j| = 1 and [[rho].sub.ij] = 0 for |i - j| > 1, where 1 [less than or equal to] i, j [less than or equal to] M. Freeman and Kuzmack  tabulated percentage points g, to 2D, for the same correlation structure for M = 5(2)9, 9(5)29, p = 0.90, 0.95, 0.99 and v = (M + 1)k for k = 9, 19, 49, 99, 499, using Monte Carlo sampling.

Dunn, Kronmal, and Yee  computed, using Monte Carlo sampling, probabilities Pr{[max.sub.1[less than or equal to]i[less than or equal to]M] |[T.sub.i]| [less than or equal to] h} = p, to 4D, for M = 2(2)20, [rho] = 0.0(0.1)0.9, h = 0.2(0.2)6.0, and v = 4(2)12, 12(4)24, 30, [infinity].

For the bivariate case M = 2, Krishnaiah, Armitage, and Breiter  tabulated probabilities Pr{[max.sub.1[less than or equal to]i[less than or equal to]M][T.sub.i] [less than or equal to] g} = p, to 6D, for [+ or -][rho] = 0.0(0.1)0.9, g = 1.0(0.1)5.5, and v = 5(1)35. Also for M = 2, Krishnaiah, Armitage, and Breiter  tabulated probabilities Pr{[max.sub.1[less than or equal to]i[less than or equal to]M]|[T.sub.i]| [less than or equal to] h} = p, to 6D, for |[rho]| = 0.0(0.1)0.9, h = 1.0(0.1)5.5, and v = 5(1)35.

Tong  tabulated percentage points g for the following correlation structure: [[rho].sub.ij] = 1 for i = j, [[rho].sub.ij] = 1/2 for i [not equal to] j and 1 [less than or equal to] i, j [less than or equal to] m or m < i, j [less than or equal to] M, [[rho].sub.ij] = -1/2 for 1 [less than or equal to] i [less than or equal to] m and m < j [less than or equal to] M or 1 [less than or equal to] j [less than or equal to] m and m < i [less than or equal to] M where m = M/2 if M is even and m = (M + 1)/2 if M is odd. His Table 1 gives g, to 7D, for M = 1(1)10, 10(2)20, p = 0.50, 0.75, 0.90, 0.95, 0.975, 0.99 and degrees of freedom v = [infinity]. His Table 2 gives g, to 5D, for M = 2(1)6, 6(2)12, 12(4)20, p = 0.50, 0.75, 0.90, 0.95, 0.975, 0.99 and degrees of freedom v = 5(1)10,10(2)20, 20(4)60, 60(30)120.

Trout and Chow  tabulated two-sided nonequicoordinate p X 100 percentage points of trivariate (M = 3) t-distribution with non-singular correlation matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

They tabulated d, to 2D, where

for p = 0.95, v = 5(1)9(2)29, a = 0.5(0.1)1.5, b = 0.5(0.1)1.5, and a set of 22 triplets ([[rho].sub.12], [[rho].sub.13], [[rho].sub.23]), where [[rho].sub.ij] = 0.0, 0.1, 0.5, 0.9, (i [not equal to] j, 1 [less than or equal to] i, j [less than or equal to] 3).

Dutt  tabulated the probabilities Pr{[max.sub.1[less than or equal to]i[less than or equal to]M] [T.sub.i] [less than or equal to] g} = p, to 6D, for g = 0.0(0.5)2.0, 2.0(1.0)4.0, and v = 8(4)40, and [infinity]: for M = 3 with ([[rho].sub.12], [[rho].sub.13], [[rho].sub.23]) = (0.3, 0.5, 0.7), and (0.1, 0.3, 0.5); and for M = 4 with ([[rho].sub.12], [[rho].sub.13], [[rho].sub.14], [[rho].sub.23], [[rho].sub.24], [[rho].sub.34]) = (0.05, 0.10, 0.15, 0.25, 0.60, 0.80), and (0.25, 0.35, 0.50, 0.60, 0.65, 0.70).

Bechhofer and Dunnett  tabulated one-sided and two-sided upper equicoordinate percentage points for M = 2(1)16, 16(2)20, degrees of freedom v = 2(1)30, 30(5)50, 60(20)120, 200, [infinity], and [rho] = 0.0(0.1)0.9, and 1/(1 + [square root of M]). They tabulate, to 5D, g and h for p = 0.80, 0.90, 0.95, and 0.99. They also tabulate equicoordinate and non-equicoordinate onesided percentage points for block correlation structure. Bechhofer and Dunnett  summarize previous tables of percentage points for equicorrelated multivariate normal and t-distributions.

4.3 Distribution of the Wilks's Likelihood Ratio Test Statistic

Schatzoff , Pillai and Gupta , Lee , and Davis  tabulate multiplying factors C to obtain upper percentage points of the distribution of the Wilks's Likelihood Ratio Test Statistic - [n - p - (1/2) (m - r + 1)] logW from the percentage points of the chi-square distribution for multivariate analysis of variance. Muirhead  has consolidated these into one large table. Here, n is the number of multivariate measurements, p is the number of regression parameter vectors, n - p is the error degrees of freedom, m is the dimension of multivariate measurements, and r is the degrees of freedom of the general linear hypothesis. Factors for the upper [alpha] X 100 percent points are tabulated for [alpha] = 0.100, 0.050, 0.025, and 0.005. The chi-square distribution has mr degrees of freedom. The degrees of freedom n - p - m + 1 equal 1(1)10, 10(2)20, 24, 30, 40, 60, 120, and [infinity]. Pairs (m, r) are such that m = 3(1)10, 12, and r [greater than or equal to] m, where r is up to 22 for m = 3, and 4, r is up to 20 for m = 5, 6, and 7, and r is up to 18, 16, and 14 for m = 8, 9, and 10, respectively. Pairs (m, r) = (6, 11), (6, 13), and (10, 13) are excluded. For r [less than or equal to] m make the substitutions m [right arrow] r, r [right arrow] m, and n - p [right arrow] n + r - p - m.

4.4 Dirichlet Distribution--Type 1

Sobel, Uppuluri, and Frankowski  tabulated, to 10D, the incomplete Dirichlet integral of Type 1:

[I.sub.p.sup.(b)] (r, n) = [[[GAMMA](n + 1)]/[[[GAMMA].sup.b](r)[GAMMA](n - br + 1)]] X [[integral].sub.0.sup.p] ... [[integral].sub.0.sup.p] (1 - [b.summation over (i=1)] [x.sub.i])[.sup.n-br] [b.[product] (i=1)] [x.sub.i.sup.r-1]d[x.sub.i]

for p = 1/b, b = 2(1)10, r = 1(1)10, and n [greater than or equal to] br. This represents Pr{[[intersection].sub.i=1.sup.b]([X.sub.i] [less than or equal to] p)} where [X.sub.1],..., [X.sub.b] have a joint Dirichlet distribution with the specified parameters. Also tabulated, to 10D, are values of [I.sub.p.sup.(b)] (r, n) for p = 1 / j, j = b + 1(1)10, b = 1(1)10, and and r = 1(1)10. Values of n are given to, 2D, for which [I.sub.p.sup.(b)] (r, n) = M for M = 0.75, 0.90, 0.95, 0.975, 0.99, 0.999, 0.9999, p = 1/j, j = b + 1(1)20, b = 1(1)10 and r = 1(1)10. Additional tables are given for Generalized Stirling Numbers and for the sample size required for occupancy problems in multinomial distributions.

4.4 Dirichlet Distribution--Type 2

Sobel, Uppuluri, and Frankowski  tabulated the incomplete Dirichlet integrals of Type 2:

[C.sub.a.sup.(b)](r, m) = [[[GAMMA](m + br)]/[[[GAMMA].sup.b](r)[GAMMA](m)]] X [[integral].sub.0.sup.a] ... [[integral].sub.0.sup.a] [[[[product].sub.i=1.sup.b][x.sub.i.sup.r-1]d[x.sub.i]]/[(1 + [[SIGMA].sub.i=1.sup.b][x.sub.i])[.sup.m+br]]],

[D.sub.a.sup.(b)](r, m) = [[[GAMMA](m + br)]/[[[GAMMA].sup.b](r)[GAMMA](m)]] X [[integral].sub.a.sup.[infinity]] ... [[integral].sub.a.sup.[infinity]] [[[[product].sub.i=1.sup.b][x.sub.i.sup.r-1]d[x.sub.i]]/[(1 + [[SIGMA].sub.i=1.sup.b][x.sub.i])[.sup.m+br]]].

The lower tail integral [C.sub.a.sup.(b)](r, m) is tabulated, to 8D, for the parameters: {r = 1(1)10, b = 1(1)15, m = 1(1)15, a = 1(1)5, and [a.sup.-1] = 2(1)5} and {r = m, b = 1(1)10, a = 0.40(0.10)0.60, a = 0.60(0.05)0.80, and [a.sup.-1] = 3(1)10}.

The upper tail integral [D.sub.a.sup.(b)](r, m) is tabulated, to 8D or 10D, for the parameters: {r = 1(1)10, b = 1(1)15, m = 1(1)15, a = 1(1)5, and [a.sup.-1] = 2(1)5}, {r = m, b = 1(1)10, a = 3(1)10, [a.sup.-1] = 0.40(0.10)0.60, and [a.sup.-1] = 0.60(0.05)0.80}, {m = r + 1, b = 1(1)10, a = 3(1)10, [a.sup.-1] = 0.40(0.10)0.60, and [a.sup.-1] = 0.60(0.05)0.80}, {r = m, b = 1(1)10, a = 0.40(0.10)0.60, a = 0.60(0.05)0.80, and [a.sup.-1] = 3(1)10}, {m = r + 1, b = 1(1)10, a = 40(0.10)0.60, 0.60(0.05)0.80, and [a.sup.-1] = 3(1)10}, {m = r + 1, r = 1(1)200, a = 1, and b = 1(1)10}, and {m = r + 2, r = 1(1)200, a = 1, and b = 1(1)10}. Values of a for which [D.sub.a.sup.(b)](r, m) = [delta] are tabulated for [delta] = 0.75, 0.95, 0.975, 0.99, 0.995, 0.999, r = 1(1)50, and b = 1(1)10. A table for expected waiting time in multinomial problems is also given.

4.6 Zonal Polynomials

Probability density functions and moments of many multivariate distributions can be evaluated using zonal polynomials. Parkhurst and James  tabulate zonal polynomials of order 1 through 12 in terms of sums of powers and in terms of elementary symmetric functions.

4.7 Distributions of the Largest and Smallest Eigenvalues of Matrices of Sample Quantities

Heck  charts some upper percentage points of the distribution of the largest eigenvalue of certain matrices of sample quantities from multivariate normal distribution. Edelman  tabulates expected values of the smallest eigenvalue of random matrices of Wishart type.

5. Summary

This article is a survey of the tables of probability distributions published about or after the publication in 1964 of the Handbook of Mathematical Functions, edited by Abramowitz and Stegun. The abstracts presented here have been verified from the original sources. Many of the distributions referenced here are implemented in commercial or publicly-available software systems.

Acknowledgment

The Handbook of Mathematical Functions, edited by Abramowitz and Stegun, is becoming increasingly out-of-date. A project is underway at the National Institute of Standards and Technology to develop a Web based replacement, a Digital Library of Mathematical Functions (DLMF). This survey of tables of probability distributions was done as a part of gathering background information for the DLMF. Comments by Ron Boisvert and Walter Liggett on an earlier draft have improved the paper. The following provided help with the DLMF and LATEX: Dan Lozier, Bruce Miller, Joyce Conlon, and Charles Hagwood.

Accepted: January 19, 2005

Available online: http://www.nist.gov/jres

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Raghu Kacker

National Institute of Standards and Technology, Gaithersburg, MD 20899-8910

and

Ingram Olkin

Stanford University, Stanford, CA 94305-4065

raghu.kacker@nist.gov

About the authors: Dr. Raghu Kacker is a mathematical statistician in the Mathematical and Computational Sciences Division of the NIST Information Technology Laboratory. Dr. Ingram Olkin is professor of statistics and professor of education at the Stanford University. The National Institaute of Standards and Technology is an agency of the Technology Administration. U.S. Department of Commerce.
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