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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 $X_2\sim\Gamma(\alpha _2,\beta _2)$?

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  • $\begingroup$ related question about Gaussians here $\endgroup$ – Glen_b Nov 14 '18 at 0:22
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You are mixing up very different things.

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

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