Markov Inequality
The Markov Inequality is a fundamental result in probability theory that provides a way to bound the probability of a non-negative random variable exceeding a certain value. Specifically, it states that for any non-negative random variable X and any positive number a, the probability that X is greater than or equal to a is at most the expected value of X divided by a. Mathematically, this is expressed as P(X \geq a) \leq \fracE[X]a.
This inequality is particularly useful because it requires minimal information about the random variable, only needing its expected value. It is often applied in various fields, including statistics and economics, to provide guarantees about the behavior of random variables without needing their entire distribution. The Markov Inequality is a key tool in understanding the concentration of measure and risk assessment.