I came across this interview question online:
Suppose you sit on the road side and observe cars driving by. Assume the distribution of cars driving by is according to an exponential distribution. Now you observe a first car after sitting for x mins, a second car after sitting for y min. Given those observations, can you estimate the parameter in the distribution?
I don't see how I am able to solve it without additional information. Thanks
By definition there should be equal probability that the observations are on either side of the mean. The mean of an exponential distribution is $\frac{1}{\lambda}$. Therefore, $\frac {x+y}{2}= \hat {(\frac{1}{\lambda})}$ is an unbiased estimator, however the inverse of this is notably a biased estimator of $\lambda$ because $E(\hat \lambda - \lambda)=\frac{\lambda}{n-1}$. So $\hat {(\frac{1}{\lambda})}\approx \frac{1}{\hat \lambda}$ giving $\hat \lambda \approx \frac {2}{x+y}$; I don't know if an unbiased estimator exists but for such a small sample-size that might be as good as you can do...