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Chebyshev's inequality

"In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean.

Specifically, no more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean

(or equivalently, at least 1 − 1/k2 of the distribution's values are within k standard deviations of the mean).

The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. "

https://en.wikipedia.org/wiki/Chebyshev%27s_inequality

"Probabilistic statement[edit]

Let X (integrable) be a random variable with finite expected value μ and finite non-zero variance σ2. Then for any real number k > 0,

{\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}\Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.

Only the case {\displaystyle k>1}k > 1 is useful. When {\displaystyle k\leq 1}{\displaystyle k\leq 1} the right-hand side {\displaystyle {\frac {1}{k^{2}}}\geq 1}{\displaystyle {\frac {1}{k^{2}}}\geq 1} and the inequality is trivial as all probabilities are ≤ 1."

"As an example, using {\displaystyle k={\sqrt {2}}} shows that the probability that values lie outside the interval {\displaystyle (\mu -{\sqrt {2}}\sigma ,\mu +{\sqrt {2}}\sigma )} does not exceed {\frac {1}{2}}."

"Markov's inequality

"Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable."

Statement[edit]

If X is a nonnegative random variable and a > 0, then the probability that X is at least a is at most the expectation of X divided by a:[1]

{\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} (X)}{a}}.}{\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} (X)}{a}}.}

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
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