MATLAB: CDF-vector- 2 random variables

cdf

I have an equation with 2 variables, x and y. x and y are random variables. The question is: Ln(Z)=aLn(x)+bLn(y)+c a,b and c are constant and already determined. I need to calculate CDF of Ln(Z). I need to obtain a surface (3D) by Matlab. I will appreciate your help.

Thanks

Best Answer

You apparently have a probability for each of the x and y values. So, you can just compute the different possible Ln(Z) values from your equation. With independence, the probability of each Ln(Z) is the product of the probabilities for the x and y used to compute it.

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MATLAB: How to plot the PDF of a random variable

From the integral that you use in order to calculate marginal PDFs, I guess that x,y are into the interval [0, +inf].

clearvars; clc; close all;
    syms x y
    Fxy(x,y) = 1-exp(-x)-exp(-y)+exp(-(x+y+x*y));    %Given CDF of X & Y
    JPDFxy(x,y) = (x+y+x*y)*exp(-(x+y+x*y));          %Calculated Joint PDF of X & Y
    MarginalPDFx(x) = int(JPDFxy,y,0,inf)          %Formula for Marginal PDF of X
    MarginalPDFy(y) = int(JPDFxy,x,0,inf)          %Formula for Marginal PDF of Y
    r=0:0.1:10; %plot PDFs for 0<x,y<10   
    pdfx=zeros(size(r)); 
    pdfy=zeros(size(r));
    for n=1:length(r)
        pdfx(n)=MarginalPDFx(r(n));
        pdfy(n)=MarginalPDFy(r(n));
    end
    figure;
    subplot(2,1,1); plot(r,pdfx,'-b'); grid on; legend('PDFx');
    subplot(2,1,2); plot(r,pdfy,'-r'); grid on; legend('PDFy');

Since Fxy is symmetrical, two PDFs are indentical:

MATLAB: Simplify(~(~(x | y) & ~(y | z)))

Remember these equivalences ( De Morgan's laws ):

 ~(x | y) <==>  ~x & ~y
 ~(x & y) <==>  ~x | ~y

Then

~(~(x | y) & ~(y | z))         ==>
~((~x & ~y) & (~y & ~z))       ==>
~(~x & ~y) | ~(~y & ~z)        ==>
~~x | ~~y | ~~y | ~~z          ==>
x | y | y | z                  ==>
x | y | z

Applying the operations from the outside to the inside is faster:

~(~(x | y) & ~(y | z))         ==>
~~(x | y) | ~~(x | z)          ==>
x | y | y | z

Best Answer

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MATLAB: How to plot the PDF of a random variable

MATLAB: Simplify(~(~(x | y) & ~(y | z)))

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