Difference of Gamma random variables

Given two independent random variables XGamma(αX,βX) and YGamma(αY,βY), what is the distribution of the difference, i.e. D=XY?

If the result is not well-known, how would I go about deriving the result?


I will outline how the problem can be approached and state
what I think the end result will be for the special case
when the shape parameters are integers, but not fill in the

  • First, note that XY takes on values in (,)
    and so fXY(z) has support (,).

  • Second, from the standard results that the
    density of the sum of two independent continuous random variables is the
    convolution of their densities, that is,
    and that the density of the random variable Y is
    fY(α)=fY(α), deduce that

  • Third, for non-negative random variables X and Y, note that the
    above expression simplifies to

  • Finally, using parametrization Γ(s,λ) to mean a
    random variable with density
    and with
    XΓ(s,λ) and YΓ(t,μ) random variables,
    we have for z>0 that
    Similarly, for z<0,

These integrals are not easy to evaluate but for the special case
s=t, Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products,
Section 3.383, lists the value of
in terms of polynomial, exponential and Bessel functions of β
and this can be used to write down explicit expressions for fXY(z).

From here on, we assume that s and t are integers so
that p(y,z) is a polynomial in y and z of degree (s+t2,s1)
and q(x,z) is a polynomial in x and z of degree (s+t2,t1).

  • For z>0, the integral (1)
    is the sum of s Gamma integrals with respect to y with coefficients
    1,z,z2,zs1. It follows that the density of
    XY is proportional to a mixture density of
    random variables for z>0. Note that this result
    will hold even if t is not an integer.

  • Similarly, for z<0,
    the density of
    XY is proportional to a mixture density of
    random variables flipped over, that is,
    it will have terms such as (μ|z|)k1exp(μz)
    instead of the usual (μz)k1exp(μz).
    Also, this result will hold even if s is not an integer.

Source : Link , Question Author : FBC , Answer Author : Dilip Sarwate

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