Probability mass function (pmf)#

Calculating the pmf from the cdf#

ctx.pmf_from_cdf(x, cdf)#

where ctx is dec, mpm, or gmp.

Returns \(\text{pmf}_X(x)\), the probability mass function, calculated from the cumulative distribution function (cdf) of a random variable \(X\):

\[\text{pmf}_X(x) = (\text{cdf}_X(x+d) - \text{cdf}_X(x))/d.\]

Using this method can be a viable option when the cdf, but not the pmf, is available in closed form.

ctx.pmf_from_cf_lattice(x, cf)#

where ctx is dec, mpm, or gmp.

Returns \(\text{pmf}_X(x)\), the probability mass function, calculated from the characteristic function \(C_X(t)\) of a random variable \(X\):

\[\text{pmf}(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-itx} C_X(t) \mathrm{d} t.\]

Using this method can be a viable option when the characteristic function, but not the pmf, is available in closed form.

ctx.pmf_from_factorialmoments(x, cf)#

where ctx is dec, mpm, or gmp.

Returns \(\text{pmf}_X(x)\), the probability mass function, of a random variable \(X\), calculated from the factorial moments \(\mu_[r]\):

\[\text{Pr}[X=x] = \sum_{j \ge x} (-1)^{x+j} \binom{j}{x} \frac{\mu'_{[j]}}{j!} = \sum_{r \ge 0} (-1)^{r} \frac{\mu'_{[x+r]}}{x!r!}\]

and

\[\sum_{i \ge x} \text{Pr}[X=i] = \sum_{j \ge x} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!}\]

Using this method can be a viable option when the factorial moments, but not the pmf, are available in closed form.

This call Edgeworth, Cornish-Fisher, or Lugganinni-Rice.