I have a random lognormally distributed variable shock defined over the range:
shock = 0.5:0.001:2;rb = 1.0204;sigma = 0.174;mu = 1;
Then I calculate a function of it:
LossF = @(LTV,shock) rb*(1 - (mu*(1 - normcdf((0.5*(sigma)^2 -log(LTV'*shock))/sigma, 0, 1))./(LTV'*shock) + 1 - logncdf(LTV'*shock, -0.5*(sigma^2), sigma))./(mu*(1 - normcdf((0.5*(sigma)^2 -log(LTV'*ones(1,size(shock,2))))/sigma, 0, 1))./(LTV'*ones(1,size(shock,2))) + 1 - logncdf(LTV'*ones(1,size(shock,2)), -0.5*(sigma^2), sigma)) );Loss = LossF(0.8,shock);
And calculate the pdf of Loss using the theorem for pdf of a function of a random variable:
pdfL = lognpdf(shock, -0.5*(sigma^2), sigma).*gradient(shock);
Checks:
When I integrate over the pdf of shock I get 1 as it should be:
trapz(shock,lognpdf(shock, -0.5*(sigma^2), sigma))
However when I integrate over the pdf of Loss i get:
trapz(Loss,pdfL) = 1.4236e-04
However2 integrating over the pdf of Loss without range gives 1:
trapz(pdfL) = 1
When I plot Loss and pdfL it seems that the range of Loss(1) and Loss(end) cover the entire range of the pdf function. How does an integral over that range return such a small number: 1.4236e-04. When integral over the same function without first argument returns 1?
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