Multidimensional Chebyshev's inequality

In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount. In probability theory, the multidimensional Chebyshev's inequality is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount. Let X be an N-dimensional random vector with expected value μ = E ⁡ [ X ] {displaystyle mu =operatorname {E} } and covariance matrix If V {displaystyle V} is a positive-definite matrix, for any real number t > 0 {displaystyle t>0} : Since V {displaystyle V} is positive-definite, so is V − 1 {displaystyle V^{-1}} . Define the random variable Since y {displaystyle y} is positive, Markov's inequality holds: