Throughout the chapter we implicitly assumed the following...

Throughout the chapter we implicitly assumed the following extension of the Chernoff bound. Prove that it is true. Let $X=\sum_{i=1}^{n} X_{i}$, where the $X_{i}$ are independent $0-1$ random variables. Let $\mu=$ $\mathbf{E}[X]$. Choose any $\mu_{L}$ and $\mu_{H}$ such that $\mu_{L} \leq \mu \leq \mu_{H}$. Then, for any $\delta>0$, $$ \operatorname{Pr}\left(X \geq(1+\delta) \mu_{H}\right) \leq\left(\frac{\mathrm{e}^{\delta}}{(1+\delta)^{(1+\delta)}}\right)^{\mu_{H}} $$ Similarly, for any $0<\delta<1$, $$ \operatorname{Pr}\left(X \leq(1-\delta) \mu_{L}\right) \leq\left(\frac{\mathrm{e}^{-\delta}}{(1-\delta)^{(1-\delta)}}\right)^{\mu_{L}} $$

Throughout the chapter we implicitly assumed the following extension of the Chernoff bound. Prove that it is true.

Let $X=\sum_{i=1}^{n} X_{i}$, where the $X_{i}$ are independent $0-1$ random variables. Let $\mu=$ $\mathbf{E}[X]$. Choose any $\mu_{L}$ and $\mu_{H}$ such that $\mu_{L} \leq \mu \leq \mu_{H}$. Then, for any $\delta>0$,
$$
\operatorname{Pr}\left(X \geq(1+\delta) \mu_{H}\right) \leq\left(\frac{\mathrm{e}^{\delta}}{(1+\delta)^{(1+\delta)}}\right)^{\mu_{H}}
$$
Similarly, for any $0<\delta<1$,
$$
\operatorname{Pr}\left(X \leq(1-\delta) \mu_{L}\right) \leq\left(\frac{\mathrm{e}^{-\delta}}{(1-\delta)^{(1-\delta)}}\right)^{\mu_{L}}
$$

Key Concepts

Moment Generating Functions

Moment generating functions (MGFs) are tools in probability theory that encode all moments of a random variable through an exponential transformation. They are especially valuable in deriving bounds such as the Chernoff bound because taking the expectation of the exponential of a random variable leverages the independence of the sum components, allowing one to obtain precise exponential estimates for tail probabilities.

Tail Probability Bounds

Tail probability bounds give estimates on the probability that a random variable deviates significantly from a reference value, typically its mean. In practice, these bounds are used to ensure that with high probability, the sum of random variables remains within a narrow band around its expected value. The extended form of the Chernoff bound discussed in the question provides such bounds even when the mean is approximated by upper and lower estimates, thereby widening the scope of its applicability.

Independent Random Variables

Independence is a fundamental property in probability theory that implies the behavior of one random variable does not affect another. This property is critical in the context of applying Chernoff bounds and concentration inequalities, as the product of the moment generating functions of individual independent variables equals the moment generating function of the sum, simplifying the task of bounding the tail probabilities.

Concentration Inequalities

Concentration inequalities are a family of results in probability theory that quantify how a random variable (often a sum of independent random variables) deviates from some central measure, usually its expected value. These inequalities, including the Chernoff bound, are crucial for understanding the stability and behavior of complex systems, offering guarantees on the likelihood of extreme outcomes.

Chernoff Bound

The Chernoff bound is a probabilistic bound that provides an exponentially decaying upper bound on the tail probabilities of sums of independent random variables. It is particularly useful for showing that such sums are tightly concentrated around their expected values, meaning the probability of large deviations decreases exponentially with the scale of the deviation.

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