Compare below: First summation: 𝑃 ( 𝑋 ≥ 𝑘 − 1 ) for X∼Binomial(n−1,p). Second summation: P(X≥k) for X∼Binomial(n,p). .
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$\begingroup$ Is it possible to say which term is grater? 𝑃 ( 𝑋 ≥ 𝑘 − 1 ) for X∼Binomial(n−1,p). P(X≥k) for X∼Binomial(n,p). $\endgroup$– KamranCommented Dec 6 at 12:31
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$\begingroup$ I believe it depends on whether $2k$ exceeds $n$ or not. Have you looked at some simple examples or attempted to analyze these sums? (They are given in closed form by standardized incomplete Beta functions.) $\endgroup$– whuber ♦Commented Dec 6 at 15:27
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