Question
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}=
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}=
Answers
There are no human answers yet.
Answered by
GPT 3.5
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: $\sigma^2 = -\frac{1}{2a}$
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: $\sigma^2 = -\frac{1}{2a}$
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.