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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kader 2015-11-26

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评论： Mayank Agrawal 2020-3-6

采纳的回答： Torsten

Hi

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Torsten 2018-6-7

The links from below are still valid.

Best wishes

Torsten.

Mayank Agrawal 2020-3-6

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

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Answer 1

Torsten 2015-11-26

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编辑：Torsten 2015-11-26

在 MATLAB Online 中打开

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

http://reference.wolfram.com/language/ref/BesselK.html

Best wishes

Torsten.

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kader 2015-11-27

Dear Torsten, I really grateful to you. Thanks a lot for this answer. In MatLab, the function is like as follows. For K1, it is besselK(1,x). Right? Though I got the expected result from the mentioned changes in MatLab. So, for K0 it will be besselK(0,x)? F(x)=1-2*sqrt((lambda1)*(lambda2)*x)*besselk(1,2*sqrt((lambda1)*(lambda2)*x))

Torsten 2015-11-27

在 MATLAB Online 中打开

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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Answer 2

Torsten 2015-11-26

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编辑：Torsten 2015-11-26

Take a look at Robert Israel's answer under

http://math.stackexchange.com/questions/729119/product-of-two-random-variables-that-has-exponential-distribution

Best wishes

Torsten.

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kader 2015-11-26

But that is for the same rate parameter for two random variable!!!

Torsten 2015-11-26

Combine answer 3 from

http://math.stackexchange.com/questions/729119/product-of-two-random-variables-that-has-exponential-distribution

and the answer from

http://math.stackexchange.com/questions/957135/product-of-two-exponentially-distributed-random-variables

and you'll arrive at the CDF.

Take the derivative to get the PDF.

Best wishes

Torsten.

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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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