Match each of the proportionality expressions below to the corresponding well-known distribution, and then compute the values of the parameter(s) of the distribution in terms of the given a, b, and/or c. The variable of interest is \theta. Express the parameters in the order of which they appear in the expression. In entering the expressions for the parameters, only the variables a, b, or c may be used.
In this problem, the distribution \textsf{N}(\mu , \sigma ^2) has parameters \mu and \sigma ^2.
\pi (\theta ) \propto c (for \theta \in [a, b] where a, b \in \mathbb {R}, a < b)
\textsf{Unif}([\alpha , \beta ])
\textsf{N}(\mu , \sigma ^2)
\textsf{Binom}(n, p)
\textsf{Beta}(\alpha , \beta )
unanswered
\text {left parameter}=
unanswered
\text {right parameter}=
unanswered
\pi (\theta ) \propto \theta ^ a(c-c\theta )^ b (for \theta \in [0, 1] where a, b > -1)
\textsf{Unif}([\alpha , \beta ])
\textsf{N}(\mu , \sigma ^2)
\textsf{Binom}(n, p)
\textsf{Beta}(\alpha , \beta )
unanswered
\text {left parameter}=
unanswered
\text {right parameter}=
unanswered
\pi (\theta ) \propto e^{a\theta ^2+b\theta +c} (for \theta \in \mathbb {R}, and it is known that a < 0)
\textsf{Unif}([\alpha , \beta ])
\textsf{N}(\mu , \sigma ^2)
\textsf{Binom}(n, p)
\textsf{Beta}(\alpha , \beta )
unanswered
\text {left parameter}=
unanswered
\text {right parameter}=
For the expression $\pi (\theta) \propto c$, the corresponding well-known distribution is a \textsf{Unif}($[a, b]$) distribution. The parameters of this distribution are:
Left parameter: $\alpha = a$
Right parameter: $\beta = b$
For the expression $\pi (\theta) \propto \theta^a (c-c\theta)^b$, the corresponding well-known distribution is a \textsf{Beta}($\alpha, \beta$) distribution. The parameters of this distribution are:
Left parameter: $\alpha = a+1$
Right parameter: $\beta = b+1$
For the expression $\pi (\theta) \propto e^{a\theta^2 + b\theta + c}$, the corresponding well-known distribution is a \textsf{N}($\mu, \sigma^2$) distribution. The parameters of this distribution are:
Left parameter: $\mu = -\frac{b}{2a}$
Right parameter: $\sigma^2 = -\frac{1}{2a}$
For the proportionality expression $\pi (\theta ) \propto c$ (for $\theta \in [a, b]$ where $a, b \in \mathbb {R}$, $a < b$), the corresponding well-known distribution is \textsf{Unif}($[\alpha , \beta ]$), and the parameters in terms of $a$, $b$, and $c$ are:
- $\text {left parameter} = a$
- $\text {right parameter} = b$
For the proportionality expression $\pi (\theta ) \propto \theta ^ a(c-c\theta )^ b$ (for $\theta \in [0, 1]$ where $a, b > -1$), the corresponding well-known distribution is \textsf{Beta}($\alpha , \beta $), and the parameters in terms of $a$, $b$, and $c$ are:
- $\text {left parameter} = a + 1$
- $\text {right parameter} = b + 1$
For the proportionality expression $\pi (\theta ) \propto e^{a\theta ^2+b\theta +c}$ (for $\theta \in \mathbb {R}$, and it is known that $a < 0$), the corresponding well-known distribution is \textsf{N}($\mu , \sigma ^2$), and the parameters in terms of $a$, $b$, and $c$ are:
- $\text {left parameter} = \mu = -\frac{b}{2a}$
- $\text {right parameter} = \sigma ^2 = -\frac{1}{2a}$