l***u
发帖数: 1 | 1 1. Fourier Inversion Theorem
Th: If \int_{-\inf}^{\inf} \abs(g_{X}(t))dt <\inf, then X is absolutely
continuous with density
f(x)=1/(2*pi) \int_{\-inf}^{\inf} exp(-itx) g(t)dt
2. The Ch. f. of Laplace distribution, which has density is 1/2*exp(-\abs(x)),
is
g(t)=1/(1+t^2)
This is easy to verify by standard calculation.
3. So, by Fourier Inversion Theorem,
1/2*exp(-\abs(x))=1/(2*pi) \int_{\-inf}^{\inf} exp(-itx)*1/(1+t^2) dt
=1/(2*pi) \int_{\-inf}^{\inf} exp(itx)*1/(1+t^2) dt
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