Distributions (polykin.distributions)

convolve_moments

convolve_moments(
    q0: float,
    q1: float,
    q2: float,
    r0: float,
    r1: float,
    r2: float,
) -> tuple[float, float, float]

Compute the first three moments of the convolution of two distributions.

If \(P = Q * R\) is the convolution of \(Q\) and \(R\), defined as:

\[ P_n = \sum_{i=0}^{n} Q_{n-i}R_{i} \]

then the first three moments of \(P\) are related to the moments of \(Q\) and \(R\) by:

\[\begin{aligned} p_0 &= q_0 r_0 \\ p_1 &= q_1 r_0 + q_0 r_1 \\ p_2 &= q_2 r_0 + 2 q_1 r_1 + q_0 r_2 \end{aligned}\]

where \(p_i\), \(q_i\) and \(r_i\) denote the \(i\)-th moments of \(P\), \(Q\) and \(R\), respectively.

PARAMETER	DESCRIPTION
q0	0-th moment of \(Q\). TYPE: float
q1	1-st moment of \(Q\). TYPE: float
q2	2-nd moment of \(Q\). TYPE: float
r0	0-th moment of \(R\). TYPE: float
r1	1-st moment of \(R\). TYPE: float
r2	2-nd moment of \(R\). TYPE: float

RETURNS	DESCRIPTION
tuple[float, float, float]	0-th, 1-st and 2-nd moments of \(P=Q*R\).

Examples:

>>> from polykin.distributions import convolve_moments
>>> convolve_moments(1., 1e2, 2e4, 1., 50., 5e4)
(1.0, 150.0, 80000.0)

Source code in src/polykin/distributions/base.py

def convolve_moments(q0: float,
                     q1: float,
                     q2: float,
                     r0: float,
                     r1: float,
                     r2: float
                     ) -> tuple[float, float, float]:
    r"""Compute the first three moments of the convolution of two distributions.

    If $P = Q * R$ is the convolution of $Q$ and $R$, defined as:

    $$ P_n = \sum_{i=0}^{n} Q_{n-i}R_{i} $$

    then the first three moments of $P$ are related to the moments of $Q$ and
    $R$ by:

    \begin{aligned}
    p_0 &= q_0 r_0 \\
    p_1 &= q_1 r_0 + q_0 r_1 \\
    p_2 &= q_2 r_0 + 2 q_1 r_1 + q_0 r_2
    \end{aligned}

    where $p_i$, $q_i$ and $r_i$ denote the $i$-th moments of $P$, $Q$ and $R$,
    respectively.    

    Parameters
    ----------
    q0 : float
        0-th moment of $Q$.
    q1 : float
        1-st moment of $Q$.
    q2 : float
        2-nd moment of $Q$.
    r0 : float
        0-th moment of $R$.
    r1 : float
        1-st moment of $R$.
    r2 : float
        2-nd moment of $R$.

    Returns
    -------
    tuple[float, float, float]
        0-th, 1-st and 2-nd moments of $P=Q*R$.

    Examples
    --------
    >>> from polykin.distributions import convolve_moments
    >>> convolve_moments(1., 1e2, 2e4, 1., 50., 5e4)
    (1.0, 150.0, 80000.0)
    """
    p0 = q0*r0
    p1 = q1*r0 + q0*r1
    p2 = q2*r0 + 2*q1*r1 + q0*r2
    return p0, p1, p2