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+ | ===3.7=== | ||

+ | Problems 3 | ||

+ | |||

+ | ===3.8=== | ||

+ | |||

+ | Problems 4 <!-- and 11--> | ||

+ | |||

+ | ===4.1=== | ||

+ | |||

+ | Problems 6 and 8 | ||

+ | |||

+ | ===4.2=== | ||

+ | |||

+ | Problem 8 | ||

+ | |||

+ | ===4.3=== | ||

+ | |||

+ | 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 $\gamma$. | ||

+ | |||

+ | ===4.6=== | ||

+ | |||

+ | Problem 5 | ||

+ | |||

+ | ===4.7=== | ||

+ | |||

+ | Problem 6 | ||

+ | |||

+ | ===Poisssson=== | ||

+ | |||

+ | Assume: | ||

+ | |||

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

+ | |||

+ | # 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. | ||

+ | # Derive the posterior distribution and its parameters in this case. | ||

+ | # Derive the marginal distribution of x. | ||

+ | # 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. | ||

+ | |||

+ | ===7.3=== | ||

+ | |||

+ | Problem 18 |