PDF of binomial distribution and mixed binomial distribution

Question

I want to plot the PDF of binomial distributed function and a mixed binomial distributed function, therefore I entered for the binomial distribution function the following:

Remove["Global`*"];
m = 45; (*number of banks*)
n = 5;(*number of defaults*)
AssetV = 100;(*asset value at T*)
DebtV = 80;(*debt value at T*)
μ = 0.2;(*annual drift*)
σ = 0.3;(*annual volatility*)
T = 5;(*years*)
ρ = 0.8; (*asset correlation*)

(*calculation of the individual default probabiltiy p^*, which is \
determined based on the Merton model*)


 P* = (CDF[
       NormalDistribution[], -(Log[
            AssetV/DebtV] + (μ - 0.5*σ^2)*T)/(σ*Sqrt[
          T])])
(*generating data table, which captures the probabilities that the \
number of defaults will be greater than n1, whereas n1 encompasses \
the range from 0 to m by the step of 1 (integer)*)

dataBD = Table[
  Probability[x >= n1, 
   x \[Distributed] BinomialDistribution[m,  P*]], {n1, 0, m,
    1}]

now I want to plot everything:

pdfBD = DiscretePlot[
  PDF[BinomialDistribution[m, P*], n1], {n1, 0, m}, 
  PlotRange -> All, AxesLabel -> {"# of Defaults", "PD"}, 
  Joined -> True, ImageSize -> 500]

so I get this plot

now I want to consider the mixed binomial distribution, determiend by the following code:

dataMBD = 
 Table[1 - 
   CDF[NormalDistribution[], ((1/
        Sqrt[ρ])*(((Sqrt[
           1 - ρ])*(InverseCDF[
            NormalDistribution[], (n1/m)]) + (InverseCDF[
           NormalDistribution[], (CDF[
             NormalDistribution[], -(Log[
                  AssetV/DebtV] + (μ - 0.5*σ^2)*
                  T)/(σ*Sqrt[T])])]))))], {n1, 0, m, 1}]

However, if I want to see its PDF and if I therefore enter this command it does not work:

DiscretePlot[PDF[dataMBD, n1], {n1, 0, m, 1}]

Is the error within the command above, or is there a fundamental problem based on wrong understanding?

hope somebody can help me. thanks.

Answer 1

First the P* that shows up in your code isn't a valid symbol, but I'll assume this is a typesetting issue or copying since you did get the first plot to work.

Probably not a complete answer, but you should look at SmoothKernelDistribution which will allow you to convert dataMBD into a distribution:

dist = SmoothKernelDistribution[dataMBD];
Plot[PDF[dist, n1], {n1, 0, 2}, PlotRange -> All]

You'll take a performance hit if you try to do something like Plot[PDF[SmoothKernelDistribution[dataMBD], n1], {n1, 0, 2}, PlotRange -> All] so define the distribution outside of Plot.