Lesson 1

Consider the following game: with probability \(2/3\) you win \(n\) rubles, with probability \(1/3\) you lose \(2n-3\) rubles. Let's compute the expected value of the winnings: \[n\cdot \frac{2}{3}+(-2n+3)\cdot \frac{1}{3}= 1 \ .\] The game appears profitable: for \(m\) repetitions, one can expect a profit of \(m\) rubles. But for which \(n\) would you agree to play even once? Would you agree for \(n=1\)? What about \(n=1\ 000\)?

As can be seen, the expected value shows around what value the random variable clusters, but it says nothing about the spread of this variable.

Variance of a random variable is responsible for its spread: the lower the variance, the more the values of the random variable are concentrated around its expected value. As in the example above, high variance is often associated with high risks.