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Integral of a PDF do not equal 1

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Hi,
I am trying to compute the integral from 0 to infinity of the following PDF:
S=@(r) 2*pi*lambdab.*r.*exp(-pi*lambdab.*r).*exp(-2*pi*lambdab.*integral(@(u) u.*exp(-pi*lambdab.*u),0,r,'ArrayValued',true));
Result=integral(S,0,inf,'ArrayValued',true);
Logically, I need to find 1 for every >0. However, when is greater than 0.2, I surprisingly get smaller values than 1 for the integral of the PDF.
Any help please!

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Accepted Answer

Steven Lord
Steven Lord on 20 May 2020
What leads you to believe the integral should be 1?
syms r u
syms lambdab positive
S=2*pi*lambdab.*r.*exp(-pi*lambdab.*r).*...
exp(-2*pi*lambdab.*int(u.*exp(-pi*lambdab.*u),u, 0,r));
int(S, r, 0, Inf)
The result I receive:
ans =
1 - exp(-2/(lambdab*pi))

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