Conditional density function with gamma and Poisson distribution It is Question 17 in Ross's book (Introduction to Probability Models-11th edition)
Let $Y$ be a gamma random variable with parameters $(s, \alpha)$. That is, its density is 
$$f_Y(y) = Ce^{-\alpha y}y^{s-1} \quad \forall\,y > 0,$$
where $C$ is a constant that does not depend on $y$. Suppose also that the conditional distribution of $X$ is given that $Y = y$ is Poisson with mean $y$. That is,
$$P(X=i\mid Y=y) = \frac{e^{-y} y ^ i}{i!} \quad \forall\, i \ne 0$$
Show that the conditional distribution of $Y$ given that $X = i$ is the gamma distribution with parameters $(s+i,\alpha+1)$.
Here is my steps:


*

*Find the joint density function


$$f_{X,Y}(i,y) = f_{X|Y}(i,y) f_Y(y) = e^{-y} y^i/i! \cdot Ce^{-\alpha y} y^{s-1} = Ce^{-(\alpha + 1)y} \cdot y^{s+i-1}/i!.$$


*Find $f_X(i)$ by
$$ \int_0^\infty f_{X,Y}(i,y) \,dy
= \int_0^\infty Ce^{-(\alpha + 1)y} \cdot y^{s+i-1}/i! \,dy ?$$

*Find $f_{Y|X}(y,i)$ by $f_{X,Y}(i,y) / f_X(i)$.
Then I am stuck at step 2, do not know how to proceed with that integral.
Thanks in advance.
 A: Just continue step (3).
\begin{align}
  f_{Y \mid X}(y,i) &= \frac{f_{X,Y}(i,y)}{f_X(i)} \\
  &= \large\frac{\frac{Ce^{-(\alpha + 1)y} \cdot
  y^{s+i-1}}{i!}}
  {\int_0^\infty \frac{Ce^{-(\alpha + 1)y} \cdot
  y^{s+i-1}}{i!} \,dy} \\
  &= \underbrace{\frac{1}{\int_0^\infty e^{-(\alpha + 1)y} \cdot
  y^{s+i-1} \,dy}}_{\text{constant independent of }y} \cdot
  e^{-(\alpha + 1)y} \cdot y^{s+i-1}
\end{align}
This is a constant multiplied by $e^{-(\alpha + 1)y} \cdot y^{s+i-1}$.
Recall that gamma distribution with parameters $(s+i,\alpha+1)$ has density $C_{(s+i,\alpha+1)} e^{-(\alpha + 1)y} \cdot y^{s+i-1}$ for all $y>0$.  By the very definition of density function,
$$\int_0^\infty C_{(s+i,\alpha+1)} e^{-(\alpha + 1)y} \cdot y^{s+i-1} \,dy = 1.$$
Observe that
$$\int_0^\infty \frac{1}{\int_0^\infty e^{-(\alpha + 1)y} \cdot
  y^{s+i-1} \,dy} \cdot
  e^{-(\alpha + 1)x} \cdot x^{s+i-1} \,dx
= \frac{\int_0^\infty e^{-(\alpha + 1)x} \cdot
  x^{s+i-1} \,dx}{\int_0^\infty e^{-(\alpha + 1)y} \cdot y^{s+i-1} \,dy}=1.$$
So $C_{(s+i,\alpha+1)} = \dfrac{1}{\int_0^\infty e^{-(\alpha + 1)y} \cdot y^{s+i-1} \,dy}$ and $f_{Y \mid X}$ is the density function of $\Gamma(s+i,\alpha+1)$.
