# Estimating a Visitation Interval: an Exercise in Bivariate Bayesian Statistics

*Loosely inspired by the German tank problem: several witnesses reported seeing a UFO during the given time intervals, and you want to quantify your certainty regarding when the UFO arrived and when it left.*

Years ago, I came up with a neat problem while teaching Bayesian statistics. Itâ€™s loosely inspired by the German tank problem.

Several witnesses reported seeing a UFO during the following time intervals:

The times represent hours in military time:

- $12$ is noon,
- $13$ is 1 pm,
- $13.5$ is 1:30 pm,
- ...

Suppose you want to quantify your certainty regarding when the UFO arrived and when it left.

Assume the data came from $\mathcal{U}[a,b],$ the uniform distribution on the interval $[a,b].$ This means the UFO arrived at time $a$ and left at time $b.$

**Watch out!** The data do NOT correspond to samples of $[a,b].$ Rather, the data correspond to subintervals of $[a,b].$

**a.** Compute the likelihood function $\mathcal{L}([a,b]\,\vert\,\text{data}).$

- Your result should come out to $\frac{3}{(b-a)^4}.$
**Hint:**if the UFO was there from $t=a$ to $t=b,$ then what's the probability that a single random observation of the UFO would take place between 12:00 and 13:30? In other words, if you had to choose a random number between $a$ and $b,$ what's the probability that your random number would be between $12$ and $13.5?$

**b.** Normalize the likelihood function so that it can be interpreted as a probability density.

- As an intermediate step to solve for the constant of normalization, you will have to take a double integral

$\begin{align*} \displaystyle \int_{b_\text{min}}^{b_\text{max}} \int_{a_\text{min}}^{a_\text{max}} \mathcal{L}([a,b]\, \vert \,\text{data}) \, \text{d}a \, \text{d}b. \end{align*}$

Two of the bounds will be infinite, and two of the bounds will be finite. - To figure out the appropriate intervals of integration for $a,b,$ ask yourself the following:
- What is the highest possible value for $a$ (i.e. the latest the UFO could have arrived) given the data?
- What is the lowest possible value for $b$ (i.e. the earliest the UFO could have left) given the data?

**c.** What is the probability that the UFO came and left sometime during the day that it was sighted? In other words, what is the probability that $0<a<a_\text{max}$ and $b_\text{min} < b < 24?$

- Your answer should come out to $\frac{41}{48}.$

**d.** What is the probability that the UFO arrived before 10am?

- Your answer should come out to $\frac{4}{9}.$