# Chebyshev's theorem

## Main Question or Discussion Point

Chebyshev's theorem: If μ and σ are the mean and standard deviation of the random variable X, then for any positive constant k,the probability that X will take on a value within k standard deviations of the mean is at least [1-(1/k²)],that is,
P(|X-μ|<kσ) ≥ 1-1/k², σ≠0.
(i) given the chebyshev theorem,prove this theorenn using classical definition of variance.
(ii)Give an example of how this theorem can be used to calculate probability.

## Answers and Replies

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Chebyshev's theorem: If μ and σ are the mean and standard deviation of the random variable X, then for any positive constant k,the probability that X will take on a value within k standard deviations of the mean is at least [1-(1/k²)],that is,
P(|X-μ|<kσ) ≥ 1-1/k², σ≠0.
(i) given the chebyshev theorem,prove this theorenn using classical definition of variance.
(ii)Give an example of how this theorem can be used to calculate probability.
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