# Numerical CDF of sum of dependent lognormal variables

Let $\left\lbrace x_1,x_2 \right\rbrace$ be a general bi-variate dependent lognormal distribution, with means $[\mu_1,\mu_2]^T$ and covariance matrix $\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2 \\ \end{pmatrix}$.

The problem is to evaluate a numerical value (analytical solution is not required and probably does not exists) of CDF of the weighted sum of these two variables, $P(w_1x_1+w_2x_2\leqslant\alpha)$, where $w_1$ and $w_2$ are real and non-negative.

Simplifying assumptions, such as $\mu_1=\mu_2$ and/or $\sigma_1=\sigma_2$ may be used, if necessary.

Solution in either Mathematica or Matlab will be appreciated.

For example,

s1=0.3;
s2=0.3;
m1 = 0;
m2 = 0;
rho=0.5;
c={{s1^2,rho s1 s2},{rho s1 s2,s2^2}}
dist = LogMultinormalDistribution[{m1, m2},   c];
Plot3D[Evaluate@{PDF[dist,{x,y}]},{x,0,3},{y,0,3},PlotRange->All]
PDF[dist, {x, y}]


shows excellent graph.

• Unfortunately, I do not know how to create the new variable $y=x_1+x_2$ by

TransformedDistribution

without using

MarginalDistribution

first, since it provides two independent distributions.

• Can you provide some more context on this or any ideas you yourself have? Jul 23, 2014 at 8:20
• Indeed, what have you tried. You do not seriously expect someone code this for you do you? Jul 23, 2014 at 8:40
• Please take a look at the update above. Sorry for not writing this at the first time - it's my first question on stackexchange ever...
– Dima
Jul 23, 2014 at 9:49

## 1 Answer

Since the sum of lognormal variables do not have closed-form expression, the proposed solution is based on the numerical evaluation of probabilities, as (following example in the question):

NProbability[z < 1,
z \[Distributed] TransformedDistribution[0.3 x + 0.4 y,
{x, y} \[Distributed]  LogMultinormalDistribution[{m1, m2}, c]]]


Moreover, the corresponding numerical CDF can be easily evaluated in this way and applied for further analysis.