For the following problem, what is the reasoning for having such prior distribution function $f_\Theta(\theta)$? If the random variable of interest $\Theta$ is the time of arrival $\theta$, I am inclined to think that $f_\Theta(\theta) = \theta$ for $0\leq\theta\leq 1$ and $f_\Theta(\theta) = 0$ otherwise. Can someone explain to me?
Which prior is "uninformative" (i.e., treats each value of $\theta$ equally)?
Because the OP seems unfamiliar with this concept, let me write a few words about the value of uninformative priors.
Suppose you choose a non-uniform prior over $\theta$, say one that makes it a higher probability for $\theta = 7$ than $\theta = 3$. What prior information do you have to assert such a case? Why is $7$ more likely than $3$ (before any data has been taken)? Suppose you somehow want to believe that $7$ is more likely than $3$ and someone else says "no... I think that $3$ is more likely than $7$." Who is right? How can you tell (before data is selected)?
The unique prior that treats all values equally is the uniform prior (here, a constant as a function of $\theta$).
The same process occurs with discrete distributions, of course. If someone gives you a six-sided die (which might be unbalanced), you don't know how it is unbalanced, i.e., which number is more likely to come up. If you magically (and with no evidence) think that $2$ is more likely to arise than $4$ (for instance), but someone else says the reverse, how could you ever demonstrate who is right (before you roll the die)? You cannot.
If you still don't understand this foundational idea in Bayesian reasoning, I urge you to look online on read any number of books on the subject, including my Pattern classification (2nd ed.) by Duda, Hart and Stork.