The proof of these statements is as follows. For formula_18 to be between "u" and "u" + "du", it is necessary that exactly "k" − 1 elements of the sample are smaller than "u", and that at least one is between "u" and "u" + d"u". The probability that more than one is in this latter interval is already formula_22, so we have to calculate the probability that exactly "k" − 1, 1 and "n" − "k" observations fall in the intervals formula_23, formula_24 and formula_25 respectively. This equals (refer to multinomial distribution for details)