Question

Let \(Y\) be a gamma random variable with parameters \((s, \alpha) .\) That is, its density is $$ f_{Y}(y)=C e^{-\alpha y} y^{s-1}, \quad y>0 $$ where \(C\) is a constant that does not depend on \(y .\) Suppose also that the conditional distribution of \(X\) given that \(Y=y\) is Poisson with mean \(y\). That is, $$ P\\{X=i \mid Y=y\\}=e^{-y} y^{i} / i !, \quad i \geqslant 0 $$ Show that the conditional distribution of \(Y\) given that \(X=i\) is the gamma distribution with parameters (s \(+i, \alpha+1\) ).

Short answer

In summary, we have shown that the conditional distribution of a gamma random variable \(Y\) given \(X = i\) follows a gamma distribution with parameters \((s+i, \alpha+1)\). We applied Bayes' theorem to find the conditional density function of \(Y\) given \(X=i\), simplified the expression, and then recognized the form of the gamma distribution. The resulting distribution's parameters were \(\alpha' = \alpha + 1\) and \(s' = s + i\), as desired.

Step-by-step solution

Apply Bayes' theorem to find the conditional density function of \(Y\) given \(X=i\).

To find the conditional density function of \(Y\) given \(X=i\), we can use Bayes' theorem. Bayes' theorem states that: $$ f_{Y|X}(y|i) = \frac{f_{Y}(y) P\\{X=i \mid Y=y\\}}{P\\{X=i\\}} $$ Substitute the given functions into the equation: $$ f_{Y|X}(y|i) = \frac{C e^{-\alpha y} y^{s-1} \cdot e^{-y} y^{i} / i !}{P\\{X=i\\}} $$

Simplify the expression.

To find the conditional density function of \(Y\) given \(X=i\), we can simplify the expression: $$ f_{Y|X}(y|i) = \frac{C e^{-\alpha y -y} y^{s-1+i} }{i ! P\\{X=i\\}} $$ Notice that the denominator \(i ! P\\{X=i\\}\) doesn't depend on \(y\), so it acts as a constant.

Recognize the gamma distribution form.

Now let's see if we can rewrite the expression such that it looks like a gamma distribution with parameters \((s+i, \alpha+1)\). Recall that the gamma distribution density function has the form: $$ f_{Y}(y)=C' e^{-\alpha' y} y^{s'-1} $$ Comparing this to our expression, we find: - \(C' = \frac{C}{i! P\\{X=i\\}}\) (constant with respect to \(y\)) - \(\alpha' = \alpha + 1\) - \(s' = s + i\)

Verify the gamma distribution parameters.

Given that we found a constant \(C'\) and \(f_{Y|X}(y|i)\) has a similar form as the gamma distribution, we can recognize that the parameters of the gamma distribution are: - \(\alpha' = \alpha + 1\) - \(s' = s + i\) Therefore, we have shown that the conditional distribution of \(Y\) given that \(X=i\) is the gamma distribution with parameters \((s+i, \alpha+1)\).