| Tchebyshev’s Inequality allows one to calculate a lower bound on the probability that a random variable will lie within a prescribed range. It is quite useful since it can be used for any distribution of values. It is:
where,
x is a random value
m is the mean
s is the standard deviation
h is a constant
Stated another way, the probability of the return lying within h standard deviations of the mean is at least: 
Remember that the Tchebyshev Inequality is valid for any distribution of random variables. However, the probability values are often overly conservative.
Roy’s Criterion
By using the Tchebyshev Inequality, we can see that Roy’s criterion holds for any distribution.
We can rewrite the inequality such that:
where,
R is the return outcome
k is a constant (a transform of h).
For Roy’s criterion, since we are only concerned with the lower limit, we can eliminate the absolute value and rewrite the term within the probability function as:
Since we can express Roy’s criterion as the number of standard deviations that k lies below the mean as:
Substituting, we get
which becomes,
While Roy’s criterion holds for any distribution of random variables, the values are likely to be very conservative. For some distributions, the values may be so conservative that they may be meaningless.
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