How to setup the problem $P(Z^4 - 17 \ge 9)$ for Exponential(4) distribution

Question

I'm working on part (d) of the problem that was discussed here on Math.StackExchange. The answer to that problem makes sense to me, and I've been able to use that answer to other, similar problems and their variations. Part (d); however, has a different setup than I've seen before, and I was hoping someone could give me a hint or push in how to approach it. :)

The problem is to compute $P(Z^4 - 17 \ge 9)$ for a $e^4$ distribution. I know from the back of the book that the answer is $e^{-25}$. I'm not sure what algebraically to do to the $Z^4 - 17 \ge 9$ part of the problem in order to set up the limits on the distribution's density function integral.

For example I tried isolating $Z^4$ and taking the 4th root, but the final $e$ term's power didn't equal $-25$.

So, in short, what is the strategy to employ on these types of problems?

Thanks!

Answer 1

From the information you've given, the answer should indeed be $$ P(Z \geq (26)^{1/4}) = e^{-4(26)^{1/4}}.$$ Are you sure you've read the question correctly? If so the answer in the book would be wrong.