Research 3 - Means and Markov Inequality

The are several kinds of mean, each with its applications and properties.
In probability and statistics, the most used kind of mean is the arithmetic mean which is defined as the central value of a discrete set of numbers and can be computed by adding all the values and then dividing the result by the total number of values. For instance, the average age is equal to the sum of the ages of every individual in a particular sample, divided by the total number of individuals. Naturally, the sample mean will differ from the population mean but, however, by the law of large numbers with a larger sample size, its mean will be closer to the population mean.

The following list summarises the other kind of means which are commonly used in statistics and mathematics.

• Weighted arithmetic mean is similar to an ordinary arithmetic mean except that, instead of each of the data points contributing equally to the final average, some data points contribute more than others.

• Geometric mean (GM) is an average that is useful for sets of positive numbers which are interpreted according to their product, instead of according to their sum.

         For instance, this mean can give a meaningful "average" when comparing individuals which have properties measured on different numeric ranges.

Suppose we've an investment which earns 10% the first year, 50% the second year, and 30% the third year. What's its average rate of return? This is not an arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied by 1.10, on second year by 1.60 and so on.

• Harmonic mean (HM) is an average which is useful for sets of numbers which are defined in relation to some unit (e.g. the speed, distance per unit of time). It can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observed variables. More formally, given n observed values, the harmonic mean is defined as follows. 

Suppose that we want to compute the average speed (km/h) of a vehicle that travel a certain distance d at a speed x of 60 km/h and then the same distance again at a speed y (40 km/h). It's average speed is the harmonic mean of x and y.

The relationship between the mentioned definition of means can be summarised by the following inequality.

AM ≥ GM ≥ HM

Naturally, the equality holds if and only if all the elements of the given sample are identical. 

Markov Inequality

This inequality gives an upper bound for the probability that a non-negative random variable is greater than or equal to some positive constant. In other words, it states that for a positive random variable X and any positive real number a, the probability that X is greater than or qual to a is less than or equal to the expected value of X divided by a.
Formally:

For instance, suppose to have a distribution X with expected value of 3. If a = 3, Markov's inequality states that P(X ≥ 3) ≤ 3/3 = 1. So this states that some value of the R.V. is greater than or equal to 3, which shouldn't be surprising. Were all the value of X less than three, then the expected value would be less than 3 as well.
As another example, suppose to being aware of the average height value of an elementary school. The Markov's inequality states that no more than 1/6 of the students can have a height greater than six times the mean height.
Finally, by considering a = cE(X), the probability that X is c times larger than it's average is at most 1/c.


One of the major use of Markov's inequality is to prove the Chebyshev's inequality, which uses the notion of variance to bound the probability that a random variable deviates far from the mean.

The aforementioned inequality follows from Markov's inequality by considering the random variable (X - E(X))² and the constant a².

*where MI indicates the use of Markov's Inequality.