Sum of gamma variables with a common scale parameter theta has distribution Gamma(sum alpha, theta) under which condition?

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Multiple Choice

Sum of gamma variables with a common scale parameter theta has distribution Gamma(sum alpha, theta) under which condition?

The sum of independent gamma variables with a common scale parameter is gamma with a shape parameter equal to the sum of the individual shapes and the same scale. This works because the moment generating function of each gamma variable with shape αi and scale θ is (1 − θt)^(−αi). Multiplying these MGFs for all summands gives (1 − θt)^(−∑αi), which matches the MGF of a Gamma(∑αi, θ). If the scale parameters differ, the product becomes ∏(1 − θi t)^(−αi), which cannot be collapsed into a single Gamma form unless all θi are identical. Hence the sum is Gamma(∑αi, θ) only when all thetas are identical.

The sum of independent gamma variables with a common scale parameter is gamma with a shape parameter equal to the sum of the individual shapes and the same scale. This works because the moment generating function of each gamma variable with shape αi and scale θ is (1 − θt)^(−αi). Multiplying these MGFs for all summands gives (1 − θt)^(−∑αi), which matches the MGF of a Gamma(∑αi, θ). If the scale parameters differ, the product becomes ∏(1 − θi t)^(−αi), which cannot be collapsed into a single Gamma form unless all θi are identical. Hence the sum is Gamma(∑αi, θ) only when all thetas are identical.

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