# Math Help - Maximum Likelihood Question

1. ## Maximum Likelihood Question

Q. A bag contains sequentially numbered lots (1,2...N). Lots are drawn at random (each lot has the same probability of being drawn). Two lots are drawn without replacement and are observed to be X_1 = 17 and X_2 = 30. What is the MLE of N, the number of lots in a bag?

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Hi everyone,
I'm really stuck on this mle chapter. Here's what I've done so far. I know it has to be the discrete uniform distribution but I'm really very stuck as to how to insert the numbers on the lots into the equation. I can't seem to find any examples like the above question.

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Use the discrete uniform distribution.
The lots are drawn without replacement, so:

P(X_1) = 1/N
P(X_2) = 1/(N-1)

There are N(N-1) possible combinations of two lots we can draw from the bag.

Hence L(N;X_i) = [(1/N)(1/(N-1))]^((N)(N-1))

l = ln L = N(N-1).ln[1/(N(N-1))]

∂l/∂N = ln[1/(N(N-1))] + (N(N-1))^2

At max, ∂l/∂N = 0

i.e. -ln[1/(N(N-1))] = (N(N-1))^2

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But here I have two problems in that the equation above is pretty horrible to be working out and also I haven't used the 30 and 17 anywhere in the equation. I know I must be wrong, but I don't know how else I can phrase the answer.

Thanks in advance for any help!

2. No need to do any calculations. The MLE is obviously $\max(X_1, X_2)$.

$L(n | x_1, x_2)
= \frac 1 n \frac 1 {n - 1} I(x_1 \le n) I(x_2 \le n) I(x_1 \ne x_2)
= \frac 1 n \frac 1 {n - 1} I[\max(x_1, x_2) \le n] I(x_1 \ne x_2)$

So, $L \downarrow$ in $n$ for $n \ge \max(x_1, x_2)$ but $L = 0$ for $n < \max(x_1, x_2)$ hence the MLE is the max.

3. Thanks! I can see how this makes mathematical sense, but I don't understand how it makes intuitive sense - the MLE is saying that, given the data, the likeliest number of lots in the bag is 30, yes? I just don't get how it makes sense intuitively...

4. Originally Posted by languagelearner23
Thanks! I can see how this makes mathematical sense, but I don't understand how it makes intuitive sense - the MLE is saying that, given the data, the likeliest number of lots in the bag is 30, yes? I just don't get how it makes sense intuitively...
It makes intuitive sense when you rememebr what the MLE is: the value of the parameter that makes the observed data most likely. In this case, it also highlights the fact that MLEs are not unbiased in general. In this instance, the MLE is highly biased.