What are alpha and beta for a uniform distribution? Are they the limits, such that f(x) is uniform between alpha < x < beta? I can't see what else then might be.
The uniform probability function f(x) is 1 for 0<x<1, and 0 elsewhere.
The meaning of a probability density function is that the probability between x and (x+dx) is f(x)dx.
Likewise if we let g(y) be the distribution of y, then the probability between y and (y+dy) is g(y)dy.
If y is a function of x, these two expressions must be equal areas.
\(\displaystyle g(y)\ dy = f(x)\ dx \implies g(y) = f(x)\left|\dfrac{dx}{dy}\right| = \frac 12 e^{-y/2},\;\;\; \text{ for } y \ge 0\)
Check if I did that right - can you now compare g(y) to a gamma distribution?
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