Integral of a PDF do not equal 1

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!

Answer 1

[Steven Lord]

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))

[Yassine Hmamouche]

many thanks!