The prior \lambda is distributed according to \textsf{Exp}(a) (a > 0). Write the probability distribution function \pi (\lambda ), in terms of \lambda and a. Do not simplify.
\pi (\lambda )=
Our expression for \pi (\lambda ) uses two variables.
Which one may be ignored (in the context of the problem) if it is used as an outermost multiplier in the expression written in proportional notation?
a
\lambda
Both a and \lambda
Write the simplified expression for \pi (\lambda ), ignoring the parameter(s) chosen earlier when used as an outside multiplier.
Hint: Your resulting expression for \pi (\lambda ) would satisfy \pi (0)=1 regardless of the value of a.
\pi (\lambda )=
Suppose that instead of it being an outermost multiplier, it plays a different role in the formula. Which of the following statements are true?
It may still be ignored if the parameter is added to the expression.
It may still be ignored if the (nonzero) parameter is divided from the expression.
It may still be ignored if the expression is taken to the power of the parameter.
If a is used as an outermost multiplier in the expression, it may still be ignored if the parameter is added to the expression.
If a is added to the expression, it may still be ignored.
If a is divided from the expression, it may still be ignored.
If the expression is taken to the power of a, it may still be ignored.
In the context of the problem, the parameter "a" can be ignored if it is used as the outermost multiplier in the expression for \pi (\lambda). Therefore, the answer to the first question is:
The parameter "a" can be ignored.
The expression for \pi (\lambda) is:
\pi (\lambda) =
Exp(a)*lambda^(a-1)
In this expression, the parameter "a" is used as the outermost multiplier, so it can be ignored.
Regarding the second question, if the parameter "a" is not used as the outermost multiplier, but instead plays a different role in the formula, the following statements are true:
1. It may still be ignored if the parameter is added to the expression.
2. It may still be ignored if the (nonzero) parameter is divided from the expression.
3. It may still be ignored if the expression is taken to the power of the parameter.
So any of these operations with the parameter "a" would allow it to be ignored.
Please note that these simplifications are specific to the context of the problem and the given expression for \pi (\lambda).