Re: Sum of gamma distributed random variables

On Mar 12, 3:45 am, Michael Haenlein <haenl...@xxxxxxxxx> wrote:
On Mar 12, 4:29 am, "danhey...@xxxxxxxxx" <danhey...@xxxxxxxxx> wrote:



On Mar 11, 3:39 pm, Michael Haenlein <haenl...@xxxxxxxxx> wrote:

Dear all,

I have three questions regarding summing up gamma distributed random
variables:

(1) Is there a closed-form expression for the sum of two gamma random
variables that have the same scale parameter but different shape
parameters? Essentially, what's the sum of a gamma (p,v) random
variable and a gamma (q,v) random variable (p being the shape
parameter and v the scale parameter)?

(2) I think to remember that the sum of n i.i.d. gamma (p, í) random
variables is itself distributed gamma with shape parameter pn and
scale parameter í. Is this correct?

(3) Another think I have in mind is that a gamma (p, í) random
variable multiplied by the scalar 1/x is itself distributed gamma with
shape parameter p and scale parameter íx. Again, is this correct?

Thanks very much for your help in advance,

Michael

(0) I assume you meant to state that the random variables are
independent; I'm making that assumption.

(1) There is a known closed-form. It can be found in the book of
distributions of Johnson and Kotz that contains the gamma dst. The
scale parameters can be different as well. A way to derive the result
is to multiply the Laplace transforms, expand the product by partial
fractions and invert by inspection. The details are intricate; I did
it with Macsyma (sp?). Maple or Mathematica are alternatives.

(2) Yes.

(3) Yes- Hide quoted text -

- Show quoted text -

Thanks very much for your answer!

Am I right in assuming that you are referring to the book titled
"Continuous univariate distributions" written by Norman L. Johnson,
Samuel Kotz and N. Balakrishnan?

I have another question in the same spirit, namely the sum of two
gammas where the shape parameters are different and the scale
parameters differ by a fixed factor. So essentially: gamma (p,v) +
gamma (q,vx) = ???. But I assume your logic works the same for this
case?

Again, thanks very much for your help?

Michael

I think that's the book I meant. It's been several years since I
looked this up, and I don't have access to a technical library now.

Yes, the logic works for that case. The scale parameters can be any
INTEGERS (I forgot to emphase that before). I don't know of partial
fraction expansions for non-integer powers.
.

Relevant Pages

  • Re: Sum of gamma distributed random variables
    ... Essentially, what's the sum of a gamma random ... with the same scale parameter, so without any calculations we can see ... variable multiplied by the scalar 1/x is itself distributed gamma with ...
    (sci.math)
  • Re: Sum of gamma distributed random variables
    ... variables that have the same scale parameter but different shape ... Essentially, what's the sum of a gamma random ... variable multiplied by the scalar 1/x is itself distributed gamma with ... I have another question in the same spirit, namely the sum of two ...
    (sci.stat.math)
  • Re: Sum of gamma distributed random variables
    ... I have three questions regarding summing up gamma distributed random ... Is there a closed-form expression for the sum of two gamma random ... If q and p are integers, you have the sum of two Erlang distributions ... with the same scale parameter, so without any calculations we can see ...
    (sci.math)
  • Re: Sum of gamma distributed random variables
    ... Essentially, what's the sum of a gamma random ... If q and p are integers, you have the sum of two Erlang distributions ... with the same scale parameter, so without any calculations we can see ... variable multiplied by the scalar 1/x is itself distributed gamma with ...
    (sci.math)
  • Re: Sum of gamma distributed random variables
    ... I have three questions regarding summing up gamma distributed random ... variables that have the same scale parameter but different shape ... variable multiplied by the scalar 1/x is itself distributed gamma with ... shape parameter p and scale parameter νx. ...
    (comp.soft-sys.matlab)
Quantcast