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Tristian Velazquez

Tristian Velazquez

Answered question

2022-06-13

Given is a sequence a 1 , a 2 , , a n over the alphabet { 1 , 2 , , m } chosen uniformly at random among the m n possibilities. What is the expected size of the set { a 1 , a 2 , , a n }?
If m = n it seems the answer tends to ( 1 1 / e ) n as n , but I don't know why.

Answer & Explanation

pressacvt

pressacvt

Beginner2022-06-14Added 19 answers

Define the indicator random variable I i for 1 i m as 1 if alphabet i is present in the set a 1 , , a n . Then the size of the set is simply i = 1 m I i . The expectation of this can be easily computed by linearity of expectation. The probability that I i equals 1 is given by 1 ( m 1 m ) n and therefore the expected size of the set is m [ 1 ( 1 1 m ) n ] . For m = n, the limiting value is indeed as you mentioned in the question.

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