If a ____ is not performing correctly, the operating system usually displays in on-screen warning about the problem so you can take corrective action.

To remove extra spaces from a text string while leaving single spaces between words, use the ________ function

Fill in the blank(s) with the appropriate word(s).

Given K equally sized clusters, the probability that a randomly chosen initial centroid will come from any given cluster is 1/K, but the probability that each cluster will have exactly one initial centroid is much lower. (It should be clear that having one initial centroid in each cluster is a good starting situation for K-means.) In general, if there are K clusters and each cluster has n points, then the probability, p, of selecting in a sample of size K one initial centroid from each cluster is given by Equation 8.1. (This assumes sampling with replacement.) From this formula we can calculate, for example, that the chance of having one initial centroid from each of four clusters is 4!/44 = 0.0938.

(a) Plot the probability of obtaining one point from each cluster in a sample

of size K for values of K between 2 and 100.

For K clusters, K = 10, 100, and 1000, find the probability that a

sample of size 2K contains at least one point from each cluster. You can

use either mathematical methods or statistical simulation to determine

the answer.