on October 30, 2006

i recently saw a question which says , you are given two games with following properties.

Game A: We win \$2 with probability 2/3 and lose \$1 with probability 1/3.
Game B: We win \$1002 with probability 2/3 and lose \$2001 with probability 1/3.

Which game is better financially? We have the same probability, 2/3, of winning each game, but that does not tell the whole story. What about the expected return for each game? Let random variables A and B be the payoffs for the two games. For example, A is 2 with probability 2/3 and 1 with probability 1/3. We can compute the expected payoff for each game as follows:

E [A] = 2 . 2/3 + (-1) . 1/3 = 1
E [B] = 1002 . 2/3 + (-2001) . 1/3 = 1

The expected payoff is the same for both games, but they are obviously very different! This difference is not apparent in their expected value, but is captured by variance. We can compute the Var [A] by working “from the inside out” as follows:

A - E[A] = 1 with probability 2/3
-2 with probability 1/3

(A - E[A])^2 = 1 with probability 2/3
4 with probability 1/3

E[(A - E[A])^2] = 1. 2/3 + 4 . 1/3

Var[A] = 2

Similarly, we have for Var [B]:

B - E[B] = 1001 with probability 2/3
-2002 with probability 1/3

(B - E[B])^2 = 1002001 with probability 2/3

4008004 with probability 1/3

E[(B - E[B])^2] = 1002001 . 2/3 + 4008004 . 1/3

Var[B] = 2004002

The variance of Game A is 2 and the variance of Game B is more than two million! Intuitively,
this means that the payoff in Game A is usually close to the expected value of \$1, but the payoff in Game B can deviate very far from this expected value. High variance is often associated with high risk. For example, in ten rounds of Game A, we expect to make \$10, but could conceivably lose \$10 instead. On the other hand, in ten rounds of game B, we also expect to make \$10, but could actually lose more than \$20,000!

i was amazed to see that that the topics like variance which i wondered when to use or how they are really applicable to real life , help us to decide some of practical things in life !!!