Is there an analytical answer for the mode of the convolution of two gamma signals I'm looking for the closed form answer for the mode and/or the mean of the convolution of two gamma signals. (Or alternatively rephrased as the sum of two gamma random variables (rv)).
The mode of a gamma rv is well known (Wikipedia). Looking for the mode of that of the sum of two gamma rv's with both different shape and scale parameters.
Alternatively, if someone knows an analytic expression for the mode for the convolution of two weibull, log-logistic, beta-prime, fischer-F, I'd be grateful to accept them as well. I'm not particularly married to the gamma, but any distribution of that look with control on shape/scale.
Thanks
 A: Let $X_1 \sim Gamma(shape = \alpha_1, rate = \lambda_1)$ and, independently
$X_2 \sim Gamma(shape = \alpha_2, rate = \lambda_2)$.  Then
$E(X_i) = \alpha_i/\lambda_i$ and $V(X_i) = \alpha_i/\lambda_i^2.$
So it follows immediately that 
$E(X_1 + X_2) = \alpha_1/\lambda_1 + \alpha_2/\lambda_2.$
For independent $X_1$ and $X_2,$ the variances also add.
Furthermore, if the rates are the same ($\lambda_1 = \lambda_2 = \lambda$),
then $X_1 + X_2 \sim Gamma(\alpha_1 + \alpha_2, \lambda)$ with
$E(X_1 + X_2) = (\alpha_1 + \alpha_2)/\lambda.$ (In this special
case, you could use the Wikipedia formula to find the mode.)
The situation where the rates differ is more difficult, and the
sum is not Gamma distributed.
The median of $S = X_1 + X_2$ is easily approximated by
simulation, but not necessarily easily evaluated exactly by mathematical formulas. The mode can be approximated using
density estimation on the simulated values of $S.$
For the specific case with $\alpha_1 = 3, \lambda_1 = 1$ and
$\alpha_2 = 5, \lambda_2 = 2,$ the R code below illustrates the formula for the mean and shows
methods of approximation for the median and the mode.
 alp.1 = 3; lam.1=1; alp.2 = 5; lam.2 = 2
 m = 10^5;  x1 = rgamma(m, alp.1, lam.1);  x2 = rgamma(m, alp.2, lam.2)
 s = x1 + x2;  mean(s);  median(s)
 ## 5.502104  # approx. E(S)
 ## 5.23841   # approx. median of S
 alp.1/lam.1 + alp.2/lam.2
 ## 5.5       # exact E(S)

From the histogram of 100,000 simulated values of $S$ below, it seems
that its approximate mode is a little more than 4.5, The green curve shows
a density estimator of the distribution of $S$ and the computations
below show where the density estimator reaches its maximum.

 hist(s, prob=T, col="wheat")
 lines(density(s), lwd=2, col="darkgreen")
 den = density(s)   # 'den' has x and y coordinates
 den$x[den$y==max(den$y)]
 ## 4.651678        # approximate mode of S

As here, it is typical of such a right-skewed distribution that the mean
exceeds the mode and the median is somewhere between.
