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I have a question related to Binomial distribution. Can someone help me?

Consider the following complementary CDF of a binomial distribution with parameters n and p :

$$ \bar{F}(x) = \sum_{k=x+1}^{n}(^n _k)p^k(1-p)^{n-k} $$

Is the complementary CDF non-decreasing in n? If yes, how does one prove it?

My intuition is that the complementary CDF is non-decreasing in n. But I am not sure how to prove it.

Sreekz
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1 Answers1

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Is the complementary CDF non-decreasing in n

$n$ is a given number in a Binomial law. $k$ is the variable.

It is non- increasing.

A property of any CDF is that it is non-decreasing, and it is evident as, by definition, the CDF is the cumulative probability function. Its complement, 1-CDF, is obviously non-increasing in $k$.

tommik
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  • I am interested in the case where x and p are fixed and n is varying. Consider the example where x =10, p=0.85, and n varies from 100 to 120. For this example, I want to prove mathematically that complementary CDF is non-decreasing as n varies from 100 to 120. – Sreekz Dec 04 '20 at 13:57
  • @Sreekz : in the situation you explained then the rv is not a binomial. It is a Negative Binomial, its pmf is different with respect the one you showed but the proof is the same. Any CDF is non -decreasing by definition (being a Cumulated probability function) and thus its complement is non increasing – tommik Dec 04 '20 at 14:07