Re: random variable

*To*: mathgroup at smc.vnet.net*Subject*: [mg102854] Re: [mg102833] random variable*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Sat, 29 Aug 2009 06:33:49 -0400 (EDT)*Reply-to*: hanlonr at cox.net

Clear[pdf, cdf, m, s, b, n1, n2]; Define some distribution such as pdf[b_, n1_, n2_, x_] := (n2*b^(-n1/n2))/Gamma[n1/n2]* x^(n1 - 1)*Exp[-x^n2/b] Check the pdf Assuming[{b > 0, n1 > 0, n2 > 0}, Integrate[pdf[b, n1, n2, x], {x, 0, Infinity}]] 1 Define the cdf (this might have to be done numerically) cdf[b_, n1_, n2_, x_] = Assuming[ {n1 > 0, n2 > 0, b > 0, x >= 0}, Simplify[ Integrate[pdf[b, n1, n2, t], {t, 0, x}]]] 1 - Gamma[n1/n2, x^n2/b]/Gamma[n1/n2] Check the cdf Assuming[{b > 0, n1 > 0, n2 > 0}, Limit[cdf[b, n1, n2, x], x -> #]] & /@ {0, Infinity} {0,1} Determine the mean m[b_, n1_, n2_] = Assuming[ {n1 > 0, n2 > 0, b > 0}, Integrate[x*pdf[b, n1, n2, x], {x, 0, Infinity}]] (b^(1/n2)*Gamma[(n1 + 1)/n2])/Gamma[n1/n2] Determine the standard deviation s[b_, n1_, n2_] = Assuming[ {n1 > 0, n2 > 0, b > 0}, Simplify[Sqrt[ Integrate[(x - m[b, n1, n2])^2*pdf[b, n1, n2, x], {x, 0, Infinity}]]]] (b^(1/n2)*Sqrt[Gamma[n1/n2]*Gamma[(n1 + 2)/n2] - Gamma[(n1 + 1)/n2]^2])/Gamma[n1/n2] The inverse cdf is determined numerically icdf[b_?NumericQ, n1_?NumericQ, n2_?NumericQ, q_?NumericQ] := Module[{mu = m[b, n1, n2]}, Chop[x /. FindRoot[cdf[b, n1, n2, x] == q, {x, mu}][[1]]]] A random number drawn from the distribution is (given that it must be nonnegative for this distribution) rand[b_?NumericQ, n1_?NumericQ, n2_?NumericQ] := Module[{r = I}, While[Head[r] === Complex || r < 0, r = icdf[b, n1, n2, RandomReal[]]]; r] Defining a specific distribution {b, n1, n2} = Ceiling[#, .25] & /@ {2 RandomReal[], 4 RandomReal[], 3 RandomReal[]} {0.75,4.,1.25} Generating the data Quiet[data = Table[rand[b, n1, n2], {400}]]; Show[{ Histogram[data, Automatic, "ProbabilityDensity"], Plot[Tooltip[pdf[b, n1, n2, x]], {x, 0, m[b, n1, n2] + 3 s[b, n1, n2]}, PlotStyle -> Directive[Red, Thick], PlotRange -> All]}] Bob Hanlon ---- omar bdair <bdairmb at yahoo.com> wrote: ============= I want to ask, how can I generate a random vaiable from some probability density functions which are not well-known? I mean, if we have some pdf which is not normal, binomial, weibull, ... but the only thing I know that it is a log-concave function, then how can I generate a number of random variables?