We will use proportionality notation in order to simplify the Gamma Distribution. But first, we perform a cosmetic change of variables to avoid repetitive notation with our answer in Part I: we write our parameters instead as \lambda _0 and q_0.
From the expression for the Gamma distribution given above, remove outermost multipliers to simplify it in such a way that our expression for f(1) is e^{-\lambda _0} regardless of the value of q_0.
Use q_0 for q_0 and lambda_0 for \lambda _0.
f(x) \propto
x^{q_0-1}e^{-\lambda_0 x}
To simplify the Gamma distribution expression, we'll remove the outermost multipliers in such a way that our expression for f(1) is e^{-\lambda _0} regardless of the value of q_0.
The Gamma distribution expression is given by:
f(x) \propto x^{q_0-1} e^{-\lambda _0 x}
To remove the outermost multipliers, we'll divide both sides of the equation by the product of these multipliers:
f(x) \propto \frac{x^{q_0-1} e^{-\lambda _0 x}}{x^{q_0-1} e^{-\lambda _0 x}} = 1
Now, we have simplified the expression such that the proportionality constant is equal to 1. This means that any constant multiple of f(x) will still be proportional to f(x). Therefore, our simplified expression for the Gamma distribution is:
f(x) \propto 1
This means that the probability density function (PDF) of the Gamma distribution does not depend on the value of x. It is a constant value.
Furthermore, from this simplified expression, we can see that f(1) will also be equal to 1, regardless of the values of q_0 and lambda_0:
f(1) = 1
Therefore, our expression for f(1) is e^{-\lambda _0} regardless of the value of q_0.