The mode of a binomial B ( n , p ) distribution is equal to k = <mo f

Emanuel Keith

Emanuel Keith

Answered question

2022-06-13

The mode of a binomial B ( n , p ) distribution is equal to k = ( n + 1 ) p . I am wondering how to estimate the probability at this value, i.e. ( n k ) p k ( 1 p ) n k , like what's the order of this value in terms of n. Is it exp( Ω ( n ))?

Answer & Explanation

Anika Stevenson

Anika Stevenson

Beginner2022-06-14Added 19 answers

The mode behaves asymptotically like n p.
You can use Stirling's approximation to show that
( n n p ) 2 n H ( p ) ( 2 π n p ( 1 p ) ) 1 / 2
where H ( p ) := p log 2 p ( 1 p ) log 2 ( 1 p ). So,
( n n p ) p n p ( 1 p ) n n p ( 2 π n p ( 1 p ) ) 1 / 2 = Θ ( n 1 / 2 ) .
In the case In the case p = 1 / 2, the above yields the approximation, the above yields the approximation π 2 n .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?