Problems 3

### 3.8

Problems 4 <!– and 11–>

Problems 6 and 8

Problem 8

Problems 1 and 6

### 4.4

Problem 4

#### Calculating Moments

$f(x; \alpha, \beta, \gamma) = \begin{cases} c & \mathrm{if}\, \alpha < x \le \beta, \\ 2c & \mathrm{if}\, \beta < x \le \gamma, \\ 0 & \mathrm{otherwise}. \end{cases}$

Calculate $c$ (the constant of integration), and then calculate the first, second, and third moments about zero, in terms of $\alpha$, $\beta$, and $\gamma$.

Problem 5

Problem 6

### Poisssson

Assume:

$x | \theta \sim Poisson(\theta)$

1. What distribution would make a good (that is, derive the conjugate) prior? Show how you came to this conclusion, that is, do not just look it up.
2. Derive the posterior distribution and its parameters in this case.
3. Derive the marginal distribution of x.
4. Using a prior with $\alpha$ (also called the shape) =5 and $\beta$ (also called the inverse scale) =5 and and data x=5. Compute the posterior.

Problem 18