# Math Help - Maximum Likelihood Estimate

1. ## Maximum Likelihood Estimate

I need help with this question :

Suppose a town has bicycles with license numbers 1, . . . , N . You observe
5 bicycles and the highest number you observe is 60. What is the MAximum Likelihood Estimate(MLE) for the number of bicycles in the town? Does the MLE provide a reasonable estimate?

2. Originally Posted by dagmary
I need help with this question :

Suppose a town has bicycles with license numbers 1, . . . , N . You observe
5 bicycles and the highest number you observe is 60. What is the MAximum Likelihood Estimate(MLE) for the number of bicycles in the town? Does the MLE provide a reasonable estimate?
Let $x_1, \, x_2, \, x_3, \, x_4, \, x_5$ be a random sample of five observations from a discrete uniform distribution with pdf $f(x_i) = \frac{1}{N}$. The largest observation is 60 (that is, $x_{(5)} = 60$).

Then $L = f(x_1, \, x_2, \, x_3, \, x_4, \, x_5) = f(x_1) \, f(x_2) \, f(x_3) \, f(x_4) \, f(x_5) = \frac{1}{N^5}$.

Note that L is a monotonically decreasing function of N and so nowhere in the interval $0 < N < \infty$ is $\frac{dL}{dN}$ equal to zero. However, note that:

1. L increases as N decreases, and

2. N must be equal to or greater than 60.

Therefore the value of N that maximises L is N = 60 (that is, $N = x_{(5)}$).

Note: This estimator is not an unbiased estimator of N.