How can I Calculate the PDF and CDF of a product of two i.i.d exponentially distributed random variables with mean a and b respectively
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Hi
3 Comments
Mayank Agrawal
on 6 Mar 2020
hello sir
I have a similar question but a little advanced one where I want to plot pdf and cdf of Z = (x*y)/(y+b) where 'b' is a positive real constant and x & y have same exponential distributions.
Can you please help me out how to plot the pdf and cdf of 'z' in MATLAB?
CDF of Z: P(Z<=z) = 1 - (e^(-x/lambda1)/lambda2)*(sqrt(4*b*z*lambda2/lambda1))*K1(sqrt(4*b*z/(lambda1*lambda2)))
so we differentiated this cdf w.r.t. z using chain rule and found the pdf which had 3 terms in addition.
we are not able to plot the pdf and cdf of z in MATLAB. please do help us out.
Thanks
Accepted Answer
Torsten
on 26 Nov 2015
Edited: Torsten
on 26 Nov 2015
You get
F(x)=1-2*sqrt((lambda1)*(lambda2)*x)*K1(2*sqrt((lambda1)*(lambda2)*x))
f(x)=2*(lambda1)*(lambda2)*K0(2*sqrt((lambda1)*(lambda2)*x))
with
lambda1 = 1/m
lambda2 = 1/n
K0, K1 : Modified Bessel functions of the second kind
Best wishes
Torsten.
2 Comments
Torsten
on 27 Nov 2015
Yes, in MATLAB notation it's
f=@(x)2*(lambda1)*(lambda2)*besselk(0,2*sqrt((lambda1)*(lambda2)*x))
F=@(x)1-2*sqrt((lambda1)*(lambda2)*x)*besselk(1,2*sqrt((lambda1)*(lambda2)*x))
Best wishes
Torsten.
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