Tables for the Calculation of B - and Z -Distribution Functions
L. N. Bol’shev, B. V. Gladkov, M. V. Shcheglova · Theory of Probability and Its Applications · 1961
Tables for the calculation of B and Z-distribution functions $I_x (p,q)$ and $F_{2p,2q} (z)$ are given. The published tables can be used as a certain supplement to Karl Pearson’s “Tables of the Incomplete Beta-function”, Biometric Laboratory, London (1934), and permit the calculation of $I_x (p,q)$ and $F_{2p,2q} (z)$ for $q \geqq 50$ and $p \leqq q$. The error of calculation is not higher than $5 \cdot 10^{ - 5} $ (for the case $q \geqq 160$ the error does not exceed $5 \cdot 10^{ - 6} $). The paper gives regions of the parameters p and q, where Tables I, II and [3] can be used for the calculation of $I_x (p,q)$. For example let us obtain $I_{0.3} (28;73)$ and $I_{0.3} (16;85)$. The point $(p,q)$ in the first case belongs to region I and for this reason we shall use Table I. With the help of (2) and (3) \[ \begin{gathered} w^2 = \frac{1} {p} + \frac{1} {q} = 0.035714 + 0.13699 = 0.049413,w = 0.222290, \hfill \\ v = \frac{1}{w}\left(\frac{1}{p} - \frac{1}{q}\right) = \frac{{0.035714 - 0.13699}} {{0.222290}} = 0.09904, \hfill \\ u = \frac{1} {w}\log \frac{{qx}} {{p(1 - x)}} = \frac{{\log 219 - \log 196}} {{0.43429 \cdot 0.22229}} = \frac{{2.34044 - 2.29226}} {{0.09654}} = 0.49917. \hfill \\ \end{gathered} \] Then: from tables [1] we have $\Phi (u) = 0.69117$, and from Table $I:\varphi _1 (u,v) = 0.01304,\varphi _2 (u,v) = - 0.01140$. Using formula (1) we finally obtain \[ \begin{gathered} I_{0.3} (28;73) = \Phi (u) + \varphi _1 (u,v) + w^2 (u,v) \hfill \\ \phantom{I_{0.3} (28;73)} = 0.69117 + 0.01304 - 0.04941 \cdot 0.01140 = 0.70365. \hfill \\ \end{gathered} \] The exact value of $I_{0.3} (28;73)$ to five decimal places is 0.70364 (f. [6]). In the second example the parametric point $(p,q)$ is in the region II. That is why we shall use Table II for obtaining $I_{0.3} (16;85)$. With the help of (7),\[ 2q + p - 1 = 185\,{\text{and }}y = \frac{{x(2q + p - 1)}}{{2 - x}} = \frac{{0.3 \cdot 185}}{{1.7}} = 32.647. \] Then by virtue of [4] and II, $I(y,p) = 0.99954$ and $\gamma (y,p) = 11$. Formula (6) gives the final value \[ 85) = I(y,p) + \frac{{\gamma (y,p)}}{{6(2q + p - 1)^2 }} = 0.99954 + \frac{{11}}{{6(185)^2 }} = 0.99959, \] the exact value of $I_{0.3} (16;85)$ to five decimal places is equal to 0.99959.