Re: Sum of gamma distributed random variables

On Mar 11, 12:40 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)?

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
that the result is Erlang with the same scale and shape = p + q. Why?
Well, (p,v) = sum of p iid exponentials with rate v and (q,v) = sum of
q iid exponentials with rate v, hence (p,v) + (q,v) = sum of (p+q) iid
exponentials with rate v. For non-integer p and q we must compute the
convolution integral, but the result is the same: gamma(p,v) +
gamma(q,v) = gamma(p+q,v).


(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?

Yes, by induction on the previous result.


(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?

Just use the fact that P{Y/x <= z} = P{Y <= xz} (for x > 0) to get the
density of Z = Y/x as (d/dz)P{Z <= z} = (d/dy)P{Y <= y}(at y = xz) * x
= x*f_Y(x*z), where f_Y(y) = gamma density. So, Yes, the result is
true.

R.G. Vickson



Thanks very much for your help in advance,

Michael

.

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