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.