# Problem from Paper 3 of the 1998 STEP; Probability distribution of Submarine call signs

#### CaptainBlack

##### Well-known member
I was browsing the 1998 STEP paper 3 (like you do) and come across a question related to a real intelligence gathering problem.

It has some simplifications and probably invalid assumptions, but is still quite interesting.

The question is:
A hostile naval power possesses a large, unknown number $$N$$ of submarines. Interception of radio signals yields a small number $$n$$ of their identification numbers $$X_i, \ \ i = 1, 2,..., n$$, which are taken to be independent and uniformly distributed over the continuous range from $$0$$ to $$N$$. Show that $$Z_1$$ and $$Z_2$$, defined by

$$\displaystyle Z_1 = \frac{n + 1}{n} \max(X_1,X_2, ...,X_n)$$

and

$$\displaystyle Z_2 = \frac{2}{n} \sum_{i=1}^n X_i$$

both have means equal to $$N$$.

Calculate the variance of $$Z_1$$ and of $$Z_2$$. Which estimator do you prefer, and why?
I don't intend to solve this since the only question of interest I can see is: "What is the pdf of $$Z_1$$ (or the moment generating function if that is the approach you prefer) ?"

(For those interested I believe the origin of this question lies in the Korean war when the size of the production run of particular objects of interest was estimated from the serial numbers on recovered debris, it is also why it is common practice to assign quasi-random serial numbers to some types of military equipment)

(Another comment may be relevant, STEP is a pre-university exam, but the nature of many of the questions inclines me to post at least some of them in the University section of MHB)

CB

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