Solutions to Final Exam
STAT 431/531

Dennis D. Cox

Notations; some useful distributions:

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Indicator functions:

$$I_A (x) \; = \; \left\{ \begin{array}{ccl} 1 & & \mbox{if ... ...\in A \\ 0 & & \mbox{if } x \notin A . \end{array} \right.$$

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uniform(a,b) has pdf $f(x) = (b-a)^{-1} I_{(a,b)} (x)$, $a < b$.

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exponential($\mu$) has pdf $f(x) \; = \; \mu^{-1} \exp (- x /\mu) I_{(0,\infty)} (x)$, $\mu > 0$.

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Poisson($\mu$) has pmf $f(x) \; = \; \mu^x e^{- \mu}/x! I_{N} (x)$ where $N = \{ 0 , 1 , 2, \ldots \}$ is the set of nonnegative integers (natural numbers), $\mu > 0$.

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The binomial distribution $B(n,p)$ has pmf $f(x) = C(n,x) p^x (1-p)^{n-x}$ where $C(n,x) = n!/(x!(n-x)!)$ if $x = 0 , 1 , \ldots n \}$ and $C(n,x) = 0$ otherwise. Here, $n$ is a positive integer and $0 \le p \le 1$.

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Bernoulli(p) is B(1,p).

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The Gamma function satisfies

$$\alpha \Gamma(\alpha) \; = \; \Gamma(\alpha + 1) , \quad \Gamma(n+1) \; = \; n!, \; n = 0, 1, 2, \ldots .$$

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The gamma($\alpha$,$\beta$) distribution has pdf

$$f(x) \; = \; \frac{x^{\alpha -1}}{\Gamma(\alpha)\beta^{\alpha}} e^{-x/\beta} I_{(0,\infty)} (x) ,$$

for $\alpha > 0$ and $\beta > 0$.

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The $\chi^2_{\nu}$ distribution is the same as the gamma($\nu/2$, $2$) distribution.