Fall 2011 Homework 5: Question 2
Joe is waiting in continuous time for a book called The Winds of Winter to be released. Suppose that the waiting time T until news of the book's release is posted, measured in years relative to some starting point, has PDF $\frac{1}{5}e^{-t/5}$ for t > 0 (and 0 otherwise); this is known as the Exponential distribution with parameter 1/5. The news of the book's release will be posted on a certain website. Joe is not so obsessive as to check multiple times a day; instead, he checks the website once at the end of each day. Therefore, he observes the day on which the news was posted, rather than the exact time T. Let X be this measurement, where X = 0means that the news was posted within the first day (after the starting point), X = 1 means it was posted on the second day, etc. (assume that there are 365 days in a year). Find the PMF of X. Is this a distribution we have studied?
Solution: X is the floor function of 365T, where the floor function of t is the greatest integer less than or equal to t. X ~ Geom(1 − e^-1/1825). Miracle check: A Geometric distribution is plausible for a waiting time, and does take values 0, 1, 2, …
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."