If X is a binomial variate with parameters n and p, where `0gt pgt` such that `(P(X=r))/(P(X=n-r))` is independent of n and r, then p equals.

Question

If X is a binomial variate with parameters n and p, where 0>p> such that P(X=r) P(X=n-r) is independent of n and r, then p equals. A. 1/2 B. 1/3 C. 1/4 D. none of these Select the correct answer from above options

Answer 1

Correct Answer - A We have , P ( X = r ) P ( X = n − r ) = . N c r p r ( 1 − p ) n − r . n C n − r p n − r ( 1 − p ) r P ⇒ P ( X = r ) P ( X = n − r ) = = ( 1 − p p ) n − 2 r = ( 1 − 1 p ) n − 2 r ⇒ For this to be independent of n and r, we must have 1 − 1 / p = 1 ⇒ p = 1 / 2 1 .