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The most important statistical concept is the expected value, which is most often just a fancy phrase for mean or average. The only necessary clarification is that we use “means” and “averages” for past outcomes and “expected value” for future outcomes.
For example, say you toss a coin, which can come up with either heads or tails and with equal probability. You receive $1 if the coin comes up heads and $2 if the coin comes up tails. Because you know that there is a 50% chance of $1 and a 50% chance of $2, the expected value of each coin toss is $1.50—repeated infinitely often, the mean will be exactly $1.50. Of course, exactly $1.50 will never come up—the expected value does not need to be a possible realization of a single coin toss.
Statisticians have invented the concept of random variables to make it easier to work with uncertainty. A random variable is a variable whose value (i.e., outcome) has not yet been determined. In the coin toss example, we can define a random variable named c (for “coin toss outcome”) that takes the value $1 with 50% probability and the value $2 with 50% probability. The expected value of c is $1.50. To distinguish a random variable from an ordinary non-random variable, we use a tilde over the variable. To denote the expected value, we use the notation E . So, in this bet, E ( c ) = 50% · $1 + 50% · $2 = $1.50 Expected Value(of Coin Toss) = Prob ( Heads ) · $1 + Prob ( Tails ) · $2 .
After the coin has been tossed, the actual outcome c could, e.g., be c = $2 , and c is no longer a random variable. Also, if you are certain about the outcome, perhaps because there is only one possible outcome, then the actual realization and the expected value are the same. The random variable is then really just an ordinary non-random variable. Is the expected outcome of the coin toss a random variable? No: we know the expected outcome is $1.50 even before we toss the coin. The expected value is known, the uncertain outcome is not. The expected value is an ordinary non-random variable; the outcome is a random variable. Is the outcome of the coin throw after it has come down heads a random variable? No: we know what it is (heads), so it is not a random variable.