Mixed Distribution of zeros and a lognormal dist.

Questions
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1

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]

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2

One idea for generating a distribution with all of its mass at zero is

EmpiricalDistribution[{0.0,0,0}]
3

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] // InputForm

Out ==>

Piecewise[{{1/20, x == 0}, {1/20 + (19*Erfc[(5*(1/10 - Log[x]))/Sqrt[2]])/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]

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4

Thanks so much! Very helpful indeed.