If 1 ≤ α show show that the gamma density has a maximum at α...
If show show that the gamma density has a maximum at .
So using this form of the gamma density function:
I would like to maximize this. Now i was thinking of trying to differentiate with respect to t, but that is becoming an issue because if i remember correctly the denominator is not a close form when the bound goes to infiniti
Answer & Explanation
The denominator is a constant as you are integrating it over t. So it is same as maximizing only the numerator. And .
Equating this to 0 you get
Check the double derivative at these points to get the maximum.
You're trying to maximize on the interval . That is the same as maximizing the logarithm of that function, since the logarithmic function is increasing, and taking the logarithm makes it a bit simpler:
The derivative of this with respect to t is .
Since the denominator is positive, the question now is only when the numerator is positive, negative, and zero. The numerator is positive if and negative if , and zero of . Hence the original function increases on and decreases on , reaching its peak at .
All this is true if ; if then the function decreases on and has its maximum at .