ASRANetBy substituting U_X-xtheaboveequation0xreduces toUU2J f(U)dUdU =2元8-8where U is called the standardised normal variateandf(U)hastheform-U222元The tabulated values of the above two functions areshown in Tables (a) and Table (b)11
By substituting the above equation reduces to where U is called the standardised normal variate and f(U) has the form The tabulated values of the above two functions are shown in Tables (a) and Table (b). dU U dU f U U U F U e 2 2 2 1 2 2 2 1 U f U e 11
ASRANetOrdinalesof theprobabilitydensiyfunction,fu(u)(1/V2r)e-s.0000.010.02-0.03t0.040.059.060.070.080.090.0039890.39890.39890 39S80.39860.39840.3982039800.39770.39730.103970.39650.39610.39560.39510.39450.39390.39320.39250.391S0.20.39100.39020.38940.38850.387603867038570.38470.38360.38250.3/0.381+0.38020.37900.37r80.37650.37520.3739037250.37120369704:0.36830.36680.36530.36370.36210.36050.35890.35720.35530.35380.5:0.35210.35030.34850.34670.34480.34290.34100.33910.33720.33520.6,0.3330.33120.32920.32710.32510.32300.32090.31870.31660.3144070.31230.31010.30790.30560.30340.30110.29890.29660.29430.29200.80.2890.28740.28500.28270.28030.27800.27560.27320.2709026S50.90.26610.26370.26130.25890.25650.2541025160.24920.24680.24441.00.24200.23960.23710.23470.23230.22990.22750.22510.22270.22031.10.21790.21550.21310.21070.20830.2059020360.20120.19890.19651.2/0,19420.19190.18950.18720.18490.13260.18040.17S10.17580.17361.30.17140.16910.16690.16260.16470.16040.15820.15610.15390.15181.4,0.14970.14760.14560.14350.14150.13940.13740.13540.13340.13131.50.12950.12760.12570.12380.12190.12000.11820.11630:11450.11271.60.11090.10920.10740.10570.10400.10230.10060.09830.097280.095661.70.094050.092460.090890.089330.087800.086280.084780.083290.081830.080381.80.078950.077340.076140.074770.073410.072060.070740.069430.068140.066871.90.065620.064380.063160.061950.060770.059590.058440.057300.056180.055082.00.053992.50.017533.00.004433.5110.0008734.00.0001345.00.00000149t TableA.1 istaken in part from,Hald (1952] and National Bureau of Standards [1953]with respectivepermissions of theauthorsandpublishers.SeeChap.3forfullreferencedetailsofall appendixreferencesTable(a).ValuesofStandardnormaliseddistribution12
Table(a). Values of Standard normalised distribution 12
ASRANetfe(u) duThe cumulative diatrtbution function, Fo(u)0.090.070.080.050.060.040.020.030.010.00140.5359523952790.33190.519900.0.51600.50800.31200.010.350000.30400.57530.56360.56750.57140.55570.55960.54780.55170.10.53980.54380.61410.61030.80260.60640.58710.59100.59480.59870.20.57930.58320.64800.65170.64060.64430.62930.6331o63680.310.62550.62170.617920.68440.68790.67720.68080.66640.67000.6736o.0.6628.40.85910.65540.72240.71907088o71230.71370.70190.70540.0.50.69850..69150.695070.75170.75490.O4540.74860.73570.738974220.60.73240.7291072570.7823O.18520.77940.78730.77030.7734077640.0.76420.75800.76110.81060.81330.80230.80510.80780.79950.79670.80.79390.78810.79100.83650.83890.82890.83150.83400.82640.90.82120.82380.81590.81860.86210.85770.85990.85080..8531085541.00.84610.84850.0.8438.84130.87900.88100.88300.87290.87490.S7700.86860.87081.110.0.866586430.901470.89800.89970.89440.39620.89070.89251.20.88490.88880.88690.916210.917740.911490.913090.914660.908240.909883O.903200.904900.906580.930560.931890.925070.926470.927850.929220.923640.9222040.919240.920730.90.942950.944080.938220.939430.940620.941790.936991.5l0.933740.933190.934480.95449O.952540.953520.950530.951540.948450.949501.610.0.94738.945200.94630O.961640.962460.963270.959070.959940.960800.958181.70.955430.957280.956370.969950.970620.967840.968560.969260.966380.967120.965621.80.964070.964850.976150.976700.974410.975000.975580.973200.973811.90.972570.971280.971932.00.97725a.10.982142.290.986102.390.989282.490.991803.724.274.753.205.616.006.7112.323.096.3625M0.9937910~110~10~10~410~10~10~1010-1110~110~11 - F(u)3.00..99865-3.50.9997674.0.0.99996834.510.99999665.00.999999715.50.999999981Table(b).Valuesof standardisednormaldistribution13
Table (b). Values of standardised normal distribution 13
ASRANetAnothertype of distribution is the log-normal (note:fora log-normal distribution - that it is unsymmetrical; thatthe mean and the median are not the same, that it islog to base e and not to base 1o). The cumulativedistribution function isgiven by+x, 11n?1dxLr-exp2212.元-αx1n(x/ μ)Puttingta1n(x/μ)theaboveequationreducesto了exp--F(x14
Another type of distribution is the log-normal (note: for a log-normal distribution - that it is unsymmetrical; that the mean and the median are not the same, that it is log to base e and not to base 10). The cumulative distribution function is given by Putting the above equation reduces to dx x x x F x 2 2 1 1 2 1n exp - 1 2 1 n x t 1 dt n x F x 1 2 t exp - 2 1 ( ) 2 14