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Let $X$ denote number of success in $n$ independent Bernoulli trials with probability of success $p$ in each trial. Show that, $$\mathbb{P}[X\ge r] \le \frac{r(1-p)}{(r-np)^2}, \quad if \quad r>np$$.

Comments- It looks like a standard problem, but I don't have enough ideas on how to proceed after opening the probability expression as sum of different combinations.

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Hint:

\begin{align} P(X \ge r) &= P(X- np \ge r-np) \\ &\le P(|X-np| \ge r-np) \\ \end{align}

  • Try to use a famous inequality. Notice that there is a square in the upperbound, it should suggest an appropriate inequality to use.
  • Also, note that $r > np$.
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  • $\begingroup$ I wonder if this trick works for a similar upper bound of $P(X\le r)$, namely $\frac{(n-r)p}{(np-k)^2}$ when $r<np$. $\endgroup$ Mar 2 '18 at 19:56
  • $\begingroup$ Let $Y$ be $X+Y=n$, then $P( X \le r) = P(Y \ge n-r)$ then we can apply the result. $\endgroup$ Mar 2 '18 at 20:10
  • $\begingroup$ Indeed, yes. But if I work out $P(Y \ge n-r)$ from scratch as you hinted in the original post, I end up with $P(Y \ge n-r)<\frac{npq}{(np-k)^2}$ when $np>k$. That is, by using Chebyshev's inequality. $\endgroup$ Mar 2 '18 at 20:50
  • $\begingroup$ $Y \sim Bin(n, 1-p)$, so I shall just replace $r$ with $n-r$ and $p$ with $1-p$ right? $\endgroup$ Mar 2 '18 at 20:57
  • $\begingroup$ Right. But what happens when you do the proof itself as you suggested in your original post? $\endgroup$ Mar 2 '18 at 20:59

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