# Why does not the weighted sum of gamma distribution come from weighted gamma variables?

If $$Z\sim 0.3\Gamma(\alpha _1,\beta _1)+0.7\Gamma (\alpha _2,\beta_2)$$, why isn't $$Z=0.3X_1+0.7X_2$$? $$X_1\sim\Gamma(\alpha _1,\beta _1)$$ and $$X_2\sim\Gamma(\alpha _2,\beta _2)$$?

• related question about Gaussians here – Glen_b Nov 14 '18 at 0:22

A probability density function like you are describing with $$P(Z) = 0.3 P(X) + 0.7 P(Y)$$ is likely a mixture distribution (see here). That means the value of Z is realized first picking whether to choose from the distribution of X (30% probability) or Y (70%) at random and then picking a value from the chosen distribution.
Secondly, if you look at sums of random variables like $$Z=X+Y$$ the resulting probability density is given as the convolution of the individual distributions. See here. Now for sums of random gamma variables the sum is a little more tricky. It is easy if you have the same scale factor, see here