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 ] and covariance matrix
V = E [ ( X − μ ) ( X − μ ) T ] . If V is a positive-definite matrix, for any real number t > 0 :
Pr ( ( X − μ ) T V − 1 ( X − μ ) > t ) ≤ N t 2 Since V is positive-definite, so is V − 1 . Define the random variable
y = ( X − μ ) T V − 1 ( X − μ ) . Since y is positive, Markov's inequality holds:
Pr ( ( X − μ ) T V − 1 ( X − μ ) > t ) = Pr ( y > t ) = Pr ( y > t 2 ) ≤ E [ y ] t 2 . Finally,
E [ y ] = E [ ( X − μ ) T V − 1 ( X − μ ) ] = E [ trace ( V − 1 ( X − μ ) ( X − μ ) T ) ] = trace ( V − 1 V ) = N . 