I understand that the following question requires converting continuous r.v. to discrete r.v. But How can we get a PMF from the CDF of continuous distribution? It involves dividing continuous values into discrete ones.. Someone please give a solution to this question with an intuition.
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 $\frac{1}{5}$ $e^{-\frac{t}{5}}$ 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. LetX be this measurement, where X = 0 means 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.
The news about the book's release could go on the website anytime. So it is a continuous random variable. Joe check's it once at the end of the day. So essentially you have to map 0<=t<=1 to X=0 (quoting the question"Let X be this measurement, where X = 0 means that the news was posted within the first day (after the starting point), X = 1 means it was posted on the second day, etc. ").