Which statement correctly describes the sum of gamma variables when the scale parameters differ?

• A. The sum is gamma
• B. The sum is not gamma
• C. The sum is Poisson
• D. The sum is Normal

Answer: (B)

The essential idea is that gamma distributions add neatly only when all scale parameters match. If you have independent gamma variables with possibly different shapes but the same scale, their sum is still gamma, with shape equal to the sum of the shapes and the common scale. When the scales differ, that convenient closure under addition fails, so the sum becomes a convolution of gamma densities rather than a single gamma distribution. In general, it is not gamma (though it can be approximated by a Normal distribution under large shape parameters).