Show that, for a random variable X with standard deviation σ[X] and any positive real number t :

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Show that, for a random variable X with standard deviation...

Show that, for a random variable $X$ with standard deviation $\sigma[X]$ and any positive real number $t$ :
(a) $\operatorname{Pr}(X-\mathbf{E}[X] \geq t \sigma[X]) \leq \frac{1}{1+t^{2}}$
(b) $\operatorname{Pr}(|X-\mathbf{E}[X]| \geq t \sigma[X]) \leq \frac{2}{1+t^{2}}$.

Key Concepts

Variance and Standard Deviation

Variance is a measure of the spread or dispersion of a random variable around its mean, and the standard deviation is the square root of the variance. These concepts quantify how much the values of the random variable deviate, on average, from the expected value. Such measures are fundamental in probability theory, particularly when establishing inequalities that provide bounds on the probabilities of large deviations from the mean.

Markov's Inequality

Markov's Inequality is a basic result that provides an upper bound on the probability that a non-negative random variable exceeds a given threshold, in terms of its expected value. It is often used as a starting point for proving more refined concentration inequalities by applying it to appropriate transformations of the original random variable, such as its square.

One-Sided Chebyshev’s Inequality (Cantelli’s Inequality)

Cantelli’s Inequality, also known as the one-sided Chebyshev inequality, refines the classical Chebyshev’s approach by giving an upper bound specifically for one tail of the distribution. It is used to bound the probability that a random variable deviates from its mean by at least a positive amount in one direction, yielding bounds like P(X – E[X] ? t?) ? 1/(1+t²). This inequality is especially useful when the concern is overestimating the risk of large deviations on one side.

Tail Bounds and Concentration Inequalities

Tail bounds are inequalities that estimate the probabilities that a random variable deviates significantly from a central value (usually the mean). Concentration inequalities, including Chebyshev’s and Cantelli’s inequalities, provide such bounds and are crucial in various fields for assessing risk, reliability, and error. They play a central role in theoretical and applied probability by ensuring that large deviations are sufficiently unlikely, thereby quantifying the concentration of the probability measure around the mean.

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