Section 2.5 - Random Variables, Mean and the Expected Value

A random variable maps outcomes from a sample space to numbers, making the outcomes easier to manipulate. For example, there isn’t much to be done with outcomes from the coin-flip sample space { H, T }, but mapping outcomes to integers with the random variable X = { (H, 0), (T,1) } allows mathematical manipulations. The mapping used depends on what’s being done. X's previous definition might be useful when counting the number of tails, while the definition X = { (H, 1), (T, -1) } is more suited for keeping track of wins and losses.

Given a random variable X over a sample space, the probability that Xequals a particular value k is the sum of the probabilities of the outcomes that X maps to k: $$Pr(X = k) = \sum_{X(o) = k} Pr(o)$$ For example, let the random variable X map the outcome i of a single die roll to i. The probability that X is even is $$Pr(even(X)) = Pr(2)+ Pr(4) + Pr(6) = 1/6 + 1/6 + 1/6 = 1/2$$ Computing the mean (or average or expected value) of a random value is a common mathematical manipulation. A random variable’s mean is the sum of the values in the range, with each value weighted by its probability: $$\sum_{k\in range(X)} kPr(X = k)$$ The mean is often represented by $\mu$. To continue the die-roll example, X's mean is
$$\begin{eqnarray} & & 1Pr(X = 1) + 2Pr(X = 2) + \cdots + 6Pr(X = 6) \\ &=& 1\cdot1/6 + 2\cdot1/6 + \cdots + 6\cdot1/6 \\ &=& (1 + 2 + \cdots + 6)1/6 \\ &=& 21\cdot1/6 \\ &=& 3.5 \end{eqnarray}$$ A random variable’s mean is not necessarily a member of its range.