It has been observed that micro-organisms occur in "clumps" on the surface of surgical instruments, and the number X of micro-organisms in a given clump is a random variable taking values 1, 2, 3, . . . , K, where K is unknown.

(a) Suppose n = 10 clumps were observed and the number of micro-organisms found were 1, 2, 1, 1, 1, 1, 3, 1, 5, 1. Find an intuitive estimate K-hat of the parameter K from the data.

(b) Sinoe '1' was observed more frequently, a probability model for the data is P[X = 1) = 9, and the remaining probability (1 -9} is uniformly distributed among the other values greater than one, that is, P(X = m} = ((41%, for :1: = 2,. . . ,R. Write down the likelihood and the log-likelihood function for the above sample data