The gamma probability function is given by fxx1ex Where are

The gamma probability function is given by f(x;,)=(x^(-1))*e^(-x/)/(^*()). Where , are constants, and,,x>0,when >1,the max value of function occurs at what x value?

Solution

since alpha and beta are constants, the denominator is a constant

Thus, to find the value of x at which the function is maximised is when :

df / dx = 0

Taking derivaitves, we get:

x ^(a -1) * e (-x*b) * (-1/b) + e^(-x/b) * (a-1) x^(a-2) = 0

Assuming x is finite, e^(-x/b) is non zero

So,

(1/b) x^(a-1) = (a-1) x^(a-2)

x = b (a-1)

x = (ab - b) is the value of x where f (x; a,b) would be maximised

Hope this helps.