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.
Let X be an N-dimensional random vector with expected value and covariance matrix
If is a positive-definite matrix, for any real number :Proof
Since is positive-definite, so is. Define the random variable
Since is positive, Markov's inequality holds:
Finally,
