Question

7. A hierarchical model or Bayesian network is a model where random variables are functions of other random variables. These models come up widely in probabilistic modeling. Here we explore the Gamma-Poisson model, $Lambda$ ~ Gamma($alpha$, $eta$) $X|Lambda$ ~ Poisson($Lambda$), where the Gamma distribution is parametrized as described in Problem 4. (a) Find the marginal probability mass function (pmf) P[X = k] = px (k). (b) Find E[X] and Var(X). (Hint: Theorems A and B of Section 4.4 might be helpful.) (c) Let $alpha$ = r for some integer r? 1 and $eta$ = $frac{1}{q}$ for 0 < q < 1. What is the marginal distribution of X in this case? (Hint: $Gamma$(k) = (k-1)! for any integer k.)

          7. A hierarchical model or Bayesian network is a model where random variables are functions of
other random variables. These models come up widely in probabilistic modeling. Here we explore
the Gamma-Poisson model,
$Lambda$ ~ Gamma($alpha$, $eta$)
$X|Lambda$ ~ Poisson($Lambda$),
where the Gamma distribution is parametrized as described in Problem 4.
(a) Find the marginal probability mass function (pmf) P[X = k] = px (k).
(b) Find E[X] and Var(X). (Hint: Theorems A and B of Section 4.4 might be helpful.)
(c) Let $alpha$ = r for some integer r? 1 and $eta$ = $frac{1}{q}$ for 0 < q < 1. What is the marginal
distribution of X in this case? (Hint: $Gamma$(k) = (k-1)! for any integer k.)

$7. A hierarchical model or Bayesian network is a model where random variables are functions of other random variables. These models come up widely in probabilistic modeling. Here we explore the Gamma-Poisson model, Lambda Gamma(alpha, eta) X|Lambda Poisson(Lambda), where the Gamma distribution is parametrized as described in Problem 4. (a) Find the marginal probability mass function (pmf) P[X = k] = px (k). (b) Find E[X] and Var(X). (Hint: Theorems A and B of Section 4.4 might be helpful.) (c) Let alpha = r for some integer r? 1 and eta = frac1q for 0 < q < 1. What is the marginal distribution of X in this case? (Hint: Gamma(k) = (k-1)! for any integer k.)$

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