Probability density function (pdf)#
Calculating the pdf from the cdf#
- ctx.pdf_from_cdf(x, cf)#
where
ctxisdec,mpm, orgmp.Returns \(\text{pdf}_X(x)\) from the cumulative distribution function (cdf) of a random variable \(X\):
\[\text{pdf}_X(x) = \frac{\mathrm{d}}{\mathrm{d}x} \text{cdf}_X(x).\]Using this method can be a viable option when the cdf, but not the pdf, is available in closed form.
Calculating the pdf from the characteristic function#
- ctx.pdf_from_cf(x, cf)#
where
ctxisdec,mpm, orgmp.Calculates the pdf as the inverse Fourier transform of its characteristic function.
The PDF of Y is the inverse Fourier transform of its characteristic function,
\[\text{pdf}_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ity} C_X(t) \mathrm{d} y = \frac{1}{\pi} \int_{0}^{\infty} \Re \left ( e^{-itx} C_X(t) \right ) \mathrm{d} t.\]where \(\Re (z)\) denotes the real part of \(z\). We also have
\[\int_{0}^{\infty} \Re \left ( e^{-itx} C_X(t) \right ) \mathrm{d} t = \int_{0}^{\infty} \Re \left ( C_X(t) \right ) \cos(t x) \mathrm{d} t + \int_{0}^{\infty} \Im \left ( C_X(t) \right ) \sin(t x) \mathrm{d} t.\]Using the right-hand side of this equation allows for efficient use of the quadrature formula of Fillon, with the half-period \(\omega = \pi/x\).