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# Mixed Distribution of zeros and a lognormal dist.

Posted 11 years ago
 A client would like to see a mixed distribution that is a blend of a lognormal dist and zeros. The number of zeros varies but with an expected value of 5% the rest of the numbers would come out of the lognormal dist. So far, he has used a degenerate normal distribution to produce an "effective zero" . But he wants to be in numerical terms. There must be a more elegant way of modelling the 5% point mass at zero in the distribution.  Here is the code that he uses. Your help is much appreciated!l = 0.05; m = 0.10; s = 0.20;dist = MixtureDistribution[{l,1 - l}, {NormalDistribution[0, 1/10000], LogNormalDistribution[m, s]}];Plot[PDF[dist, x], {x, -1, 3}, Filling -> Axis, PlotRange -> All] 3 Replies
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Posted 11 years ago
 Thanks so much!  Very helpful indeed.
Posted 11 years ago
 Try the following:l = 1/20; m = 1/10; s = 1/5;mdist = MixtureDistribution[{l, 1 - l}, {DiscreteUniformDistribution[{0, 0}], LogNormalDistribution[m, s]}];With explicit form In ==>CDF[mdist, x] // InputFormOut ==> Piecewise[{{1/20, x == 0}, {1/20 + (19*Erfc[(5*(1/10 - Log[x]))/Sqrt])/40, x > 0}}, 0]Quoting Wikipedia: Random Variable"A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive."Therefore PDF is not a defined notion for the distribution.Plot[CDF[mdist, x], {x, -1, 3}, Filling -> Axis, PlotRange -> All] Posted 11 years ago
 One idea for generating a distribution with all of its mass at zero is EmpiricalDistribution[{0.0,0,0}]