In today’s Uncertainty Wednesday we start to explore the properties of random variables. The first one we will look at is the so-called “expected value” or EV for short. The EV is simply the probability weighted average of a random variable X, i.e.
EV = ∑P(x)*x sum is for all possible values x that X can have
Note: for now we will continue to work with so called discrete random variables which have distinct values of x and where for each value of x P(x) > 0 (as opposed to continuous random variables where x can vary in infinitesimally small increments – we will get to that later).
The first thing to note about the expected value is that it doesn’t need to be a value that the random variable can take on. This is just like when you have a group of numbers and take their average, the average doesn’t have to (and often won’t be) one of the numbers.
In fact, here are the expected values from last week’s investment examples
Investment 1: 0.99 * (-1) + 0.01 * 99 = -0.99 + 0.99 = 0
Investment 2: 0.99 * (-100) + 0.01 * 9900 = -99 + 99 = 0
Investment 3: 0.99 * (-10000) + 0.01 * 990000 = -9900 + 9900 = 0
The expected value of all three investments is 0 but the only values that the random variable can take on are -1, 99 in investment 1, -100 and 9900 in investment 2 and -10000, 990000 in investment 3.
Also quite clearly the expected value being the same for all three investments (despite their massively different payouts) means that it is not an additional measurement of risk above and beyond what was provided by the entropy of the underlying probability distribution.
Now something that causes no end of confusion is that the expected value goes by lots of different names, including “mean” and “average.” The reason that’s a problem is because we also use mean and average when we talk about a bunch of outcomes that have been realized from an underlying distribution. Just looking at the numbers above, you can easily see how you might get -1, -1, -1 three times in a row if you keep making investment 1. Now the average of that is obviously -1, which is quite clearly NOT the same as the expected value (which is 0). That’s why I like to use expected value as a term for characterizing an entire random variable and reserve mean (or better yet “sample mean”) for a set of outcomes.
Later in Uncertainty Wednesday we will learn about the relationship between the sample mean and the expected value. For now please repeat after me: I will not confuse the sample mean with the expected value!
As an exercise you may want to analyze how the expected value changes when every payout is multiplied by a number and has a constant added to it (i.e. a linear transform of the payout). So define a new random variable Y, where
y = ax + b for every x from X. What is EV(Y) in terms of EV(X)?
