Intro to Statistical Learning (2nd Ed), Solution to Problem 11.9c

Q11.9c: In this exercise, we will explore the consequences of assuming that the survival times follow an exponential distribution. (a) Suppose that a survival time follows an Exp(λ) distribution, so that its density function is f(t)=λ exp(−λt) . Using the relationships provided in Exercise 8, show that S(t)=exp(−λt) . (b) Now suppose that each of n independent survival times follows an Exp(λ) distribution. Write out an expression for the likelihood function (11.13). L=∏i=1nf(yi)δiS(yi)1−δi=∏i=1nh(yi)δiS(yi) (c) Show that the maximum likelihood estimator for λ is λ^=∑i=1nδi/∑i=1nyi (d) Use your answer to (c) to derive an estimator of the mean survival time. Hint: For (d), recall that the mean of an Exp(λ) random variable is 1/λ . Download Book: https://www.statlearning.com/ Authors' Lectures (R): https://youtube.com/playlist?list=PLoROMvodv4rOzrYsAxzQyHb8n_RWNuS1e&si=NP0wJ6RjP8XkxU7y Authors' Lectures (Python): https://youtube.com/playlist?list=PLoROMvodv4rPP6braWoRt5UCXYZ71GZIQ&si=0Z8tx4xlPLEyjZ70 https://colab.research.google.com/drive/1mAK5S-EKbq1ECuqb1wv1S9jE7Bk7ZPNA?usp=sharing