## Chebyshev's Inequality

Chebyshev's inequality may be used to determine the likelihood regarding closeness of the value of a particular random variable relative to its mean.

For random variable X with mean μ and variance σ

^{2}:

Example #1:

For random variable X greater than with
a binomial
distribution with probability of success equal to 0.3 and
number of trials equal to 100, determine the upper bound regarding
probability of X
residing between 20 and 40 using Chebyshev's inequality.

Solution #1:

Given the following
values for Binomial distribution:

p = Probability of success = 0.3

N = Number of trials = 100

For Binomial distribution:

E(X) = Np = (100)(0.3) = 30

Var(X) = σ

Using Chebychev's inequality:

Pr(|X − 30| ≥ k) ≤ 21/k

Pr[(X ≥ 30 + k) or (X ≤ 30 − k)] ≤ 21/k

Using k = 10:

Pr[(X ≥ 30 + 10) or (X ≤ 30 − 10)] ≤ 21/10

Pr[(X ≥ 40) or (X ≤ 20)] ≤ 0.21

Knowing Pr[(X ≥ 40) or (X ≤ 20)] + Pr[(20 < X < 40)] = 1:

Pr[(20 < X < 40)] = 1 − Pr[(X ≥ 40) or (X ≤ 20)]

Pr[(20 < X < 40)] = 1 − 0.21

Pr[(20 < X < 40)] = 0.79

Determine the maximum probability that values of a random variable would deviate two or more standard deviations from its mean.

Pr(|X − μ| ≥ 2σ) ≤ 1/4 or

Pr(|X − μ| ≥ 2σ) ≤ 0.25

p = Probability of success = 0.3

N = Number of trials = 100

For Binomial distribution:

E(X) = Np = (100)(0.3) = 30

Var(X) = σ

^{2}= Np(1 − p) = (100)(0.3)(0.7) = 21Using Chebychev's inequality:

Pr(|X − 30| ≥ k) ≤ 21/k

^{2}Pr[(X ≥ 30 + k) or (X ≤ 30 − k)] ≤ 21/k

^{2}Using k = 10:

Pr[(X ≥ 30 + 10) or (X ≤ 30 − 10)] ≤ 21/10

^{2}Pr[(X ≥ 40) or (X ≤ 20)] ≤ 0.21

Knowing Pr[(X ≥ 40) or (X ≤ 20)] + Pr[(20 < X < 40)] = 1:

Pr[(20 < X < 40)] = 1 − Pr[(X ≥ 40) or (X ≤ 20)]

Pr[(20 < X < 40)] = 1 − 0.21

Pr[(20 < X < 40)] = 0.79

Example #2:

Determine the maximum probability that values of a random variable would deviate two or more standard deviations from its mean.

Solution #2:

Using Chebyshev's inequality with k = 2σ:

Pr(|X − μ| ≥ 2σ) ≤ σ

Pr(|X − μ| ≥ 2σ) ≤ σ

Pr(|X − μ| ≥ 2σ) ≤ σ

^{2}/(2σ)^{2}Pr(|X − μ| ≥ 2σ) ≤ σ

^{2}/4σ^{2}Pr(|X − μ| ≥ 2σ) ≤ 1/4 or

Pr(|X − μ| ≥ 2σ) ≤ 0.25

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