# Chebyshev's inequality for sum of indicator variables

If $$N = \sum_{i=0}^{n} I_i$$ where $$I_i$$ is the indicator variable of event $$A_i$$, with $$P(A_i)=p_i$$, how can one show that

$$P(N=0) \leq \frac{\operatorname{Var}(N)}{E(N)^2}$$

using Chebyshev's inequality or otherwise?

I don't know how to apply Chebyshev's inequality here – it doesn't seem like there's anything that has the right form.

Edit: The first part of the question is to calculate the expectation and variance of $$N$$ in terms of $$p_i$$ and $$p_{ij} = P(A_i \cap A_i)$$ - perhaps this could be useful for the second part?

Edit: Full question: • How do you know $p_{ij}$ without independence of events $A_i$'s? Chebyshev/Markov holds for all rv w/o looking into their characteristics, so in most cases, those bounds are no good. – GNUSupporter 8964民主女神 地下教會 Nov 27 '20 at 13:39
• We don't know what the $p_{ij}$ are, we just have to find the variance of N in terms of $p_i$ and $p_{ij}$. – asfjbkjabf Nov 27 '20 at 13:40
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• The calculations for $var(N)$ and $E(N)$ are irrelevant. Chebyshev is just Markov's inequality applied to "squared deviation from average" $(X-E[X])^2$. Markov's inequality's proof is incredibly simple: get a whole $E[X]$, truncate it with $1_{\{X \ge a\}}$, write out the inequality and bounds, and wrap out things. You may replace $E[X]$ with $var(X)$ and rework the proof. – GNUSupporter 8964民主女神 地下教會 Nov 27 '20 at 13:48

I think I have an answer to this question but would appreciate any feedback! :)

Firstly note that $$P(N=0) \leq P(|N-E(N)| = E(N))$$

We know that

$$P(|N-E(N)| \geq E(N)) \leq \frac{Var(N)}{E(N)^2}$$ using Chebyshev.

We want to connect these two expressions. We can do this by noting that $$P(|N-E(N)| \geq E(N)) = P(|N-E(N)| = E(N)) + P(|N-E(N)| > E(N)) \geq P(|N-E(N)| = E(N))$$

Therefore, putting this all together, we have

$$P(N = 0) \leq P(|N-E(N)| = E(N)) \leq P(|N-E(N)| \geq E(N))\leq \frac{Var(N)}{E(N)^2}$$

• The first equality is wrong: what about the event $\{N = 2 E[N] = 2 \sum_i p_i \}$? – GNUSupporter 8964民主女神 地下教會 Nov 27 '20 at 14:24
• Good point thanks. I have edited it – asfjbkjabf Nov 27 '20 at 14:25