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]
