If is a random variable that takes non-negative values, then the probability that is at most This simple result is called Markov's inequality. Clever uses of Markov's inequality can have surprisingly powerful consequences. The main trick is to choose an increasing function and apply Markov's inequality to the random variable instead. If is rapidly growing but is not too large, then one can prove that the probability that , which equals the probability that , is very small.
Let us begin with a proof of Markov's inequality itself. By the law of total probability we know that
Since (by hypothesis), this is at least which is obviously at least and from this it follows that , as we wanted.
It is frequently useful in probability theory (and also in probabilistic combinatorics) to prove that a random variable is close to its mean most of the time. Of course, this may not always be true, which is why we invent concepts such as variance to give us some measure of how concentrated a random variable is about its mean. A precise relationship between the variance of a random variable and the concentration of about its mean is given by Chebyshev's inequality, which states that if , then
To prove this, we first recall the definition of the variance of , which is . This instantly suggests a function that we can plug into Markov's inequality: instead of looking at the random variable we shall look at (The function is not increasing, but this is all right because we are actually interested in the random variable )
The random variable is non-negative, so by Markov's inequality we find that
and we are done, since
Suppose that we have a random variable with mean such that and Then Markov's inequality implies that while Chebyshev's inequality implies that Since all probabilities are at most 1, Markov's inequality tells us nothing unless , and Chebyshev's inequality tells us nothing unless Since this tells us that Markov's inequality will usually give us some information for smaller values than Chebyshev's inequality. However, if and are reasonably comparable, then Chebyshev's inequality is much stronger for larger values of . This illustrates a general principle: a good bound on for a quickly growing positive function gives us a good upper bound for the probability that is large. In the next example, we shall see a very strong result of this kind.
Let be independent random variables, each of mean 0 and each taking values in the interval What is the probability that ?
To get a preliminary feel for this question, let us use the fact that variances of independent events add together. Each obviously has variance at most 1, since it cannot differ from its mean by more than 1. Therefore, the variance of is at most , so by Chebyshev's inequality the probability that is at most This tells us that once gets past , the probability decays fairly rapidly. However, we shall now see that it decays far more rapidly than Chebyshev's inequality implies.
To prove this, we shall estimate the probability that and then, by symmetry, we will also have an estimate for the probability that To obtain the first estimate we shall think about the expectation of
The exponential function has three properties that make it a particularly good choice. The first is that it is non-negative. The second is that it turns sums into products, which is very useful to us because the expectation of a product of independent random variables is equal to the product of the expectations. The third is that it is a rapidly growing function.
Using these properties, we find that
We would like to show that this is not too large, and our problem is now reduced to showing that each is not too large. It is at this point that we exploit the fact that each has mean and takes values in As long as , this implies that
And now a small calculation shows that if then this is at most Indeed, we can prove this coefficient by coefficient: , and so on. Therefore,
Now we are ready to apply Markov's inequality. The probability that is equal to the probability that which by Markov's inequality is at most We are now free to choose so as to minimize this estimate, provided that The minimum occurs when when it gives This estimate is valid if , and therefore for all , since the probability that is zero.
After applying the same argument to we find that the probability that is at most so once gets beyond the probability decays very quickly indeed.
This conclusion is far stronger than that of Chebyshev's inequality, but it should be stressed that we have also used a stronger hypothesis: here we assumed that the random variables were bounded, but we could have obtained an upper bound of from the weaker assumption that each had variance at most 1.