Let X1, …, Xn be independent Poisson trials such that Pr(Xi)=pi. Let X=∑i=1^n ai Xi and μ=𝐄[X].

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Let X1, …, Xn be independent Poisson trials such that...

Let $X_{1}, \ldots, X_{n}$ be independent Poisson trials such that $\operatorname{Pr}\left(X_{i}\right)=p_{i}$. Let $X=\sum_{i=1}^{n} a_{i} X_{i}$ and $\mu=\mathbf{E}[X]$. Use the result of Exercise $4.15$ to prove that, for any $0<\delta<1$ $$ \operatorname{Pr}(|X-\mu| \geq \delta \mu) \leq 2 \mathrm{e}^{-2 \delta^{2} \mu^{2} / n} $$

Let $X_{1}, \ldots, X_{n}$ be independent Poisson trials such that $\operatorname{Pr}\left(X_{i}\right)=p_{i}$. Let $X=\sum_{i=1}^{n} a_{i} X_{i}$ and $\mu=\mathbf{E}[X]$. Use the result of Exercise $4.15$ to prove that, for any $0<\delta<1$
$$
\operatorname{Pr}(|X-\mu| \geq \delta \mu) \leq 2 \mathrm{e}^{-2 \delta^{2} \mu^{2} / n}
$$

Key Concepts

Expectation and Deviation

The expected value of a random variable is a measure of its central tendency, while deviation quantifies how much the outcomes spread out from this mean. In many probabilistic analyses, bounding the probability that a random variable deviates significantly from its expected value is crucial. This concept underlies many concentration inequalities, and it becomes especially important when working with sums of independent variables, as it provides insight into their typical behavior versus rare, extreme outcomes.

Concentration Inequalities

Concentration inequalities are mathematical tools used to bound the probability that a random variable deviates from some central value, such as its mean. These inequalities generally exhibit exponentially decaying bounds in terms of the deviation size, which is essential in quantifying the reliability of statistical estimators and algorithms. They provide theoretical guarantees that the sum of many independent random effects is tightly concentrated around its expectation.

Weighted Sum of Independent Random Variables

This concept deals with forming a new random variable by taking a linear combination of independent random variables, where each component is multiplied by a constant weight. In probability theory and statistics, analyzing such sums is central because many interesting quantities (like sample means, totals, or other aggregate measures) are expressed as weighted sums. The independence of the variables typically allows for the use of powerful techniques in evaluating their collective behavior, including variance calculation and application of concentration results.

Hoeffding's Inequality

Hoeffding's inequality is a specific concentration inequality that provides an exponential bound on the probability that the sum of bounded independent random variables deviates from its expected value. It is particularly powerful because it requires only the boundedness and independence of the random variables, not any distributional assumptions. This inequality is used in many areas, including statistical learning theory, randomized algorithms, and empirical process theory, to provide performance guarantees and confidence bounds.

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