Chebyshev's Theorem: Proving it and Explaining its Application

[risha]

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.

 

[SW VandeCarr]

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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