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 [mathjaxinline]\lambda _0[/mathjaxinline] and [mathjaxinline]q_0[/mathjaxinline].
From the expression for the Gamma distribution given above, remove outermost multipliers to simplify it in such a way that our expression for [mathjaxinline]f(1)[/mathjaxinline] is [mathjaxinline]e^{-\lambda _0}[/mathjaxinline] regardless of the value of [mathjaxinline]q_0[/mathjaxinline].
Use q_0 for [mathjaxinline]q_0[/mathjaxinline] and lambda_0 for [mathjaxinline]\lambda _0[/mathjaxinline].
[mathjaxinline]f(x) \propto[/mathjaxinline]
x^(q_0-1)*e^(-lambda_0*x)
Compare this with the posterior distribution you computed from Part I, which you should see is a Gamma distribution. What is the corresponding variable, and what are its parameters?
Use SumXi for \sum _{i=1}^ n X_ i.
x =
q_0 =
\lambda _0 =
The corresponding variable is x, and its parameters are q = q_0 + n and \lambda = \lambda_0 + \text{SumXi}.
In the expression for the posterior distribution from Part I, we had:
f(x | \text{data}) \propto x^{q} e^{-\lambda x}
Comparing this with the simplified expression for the Gamma distribution, we can see that the corresponding variable is x, and the parameters are q = q_0 - 1 and \lambda = \lambda_0.
The expression [mathjaxinline]f(x)[/mathjaxinline] represents the Gamma distribution and can be simplified as follows:
[mathjaxinline]f(x) \propto x^{(q_0-1)}e^{-\lambda_0x}[/mathjaxinline]
Comparing this with the posterior distribution computed in Part I, we can identify the corresponding variable and its parameters. The variable in this case is [mathjaxinline]x[/mathjaxinline]. The parameters for the Gamma distribution are:
[mathjaxinline]q_0[/mathjaxinline] - Shape parameter
[mathjaxinline]\lambda_0[/mathjaxinline] - Rate parameter