3.4: Random Variables

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Page ID: 47235

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Now we assign to each event $A_i$ in the sample space a given value: each $A_i$ corresponds with an $x_i$. For instance, a coin toss resulting in heads could be equated with a $1 reward, and each tails could trigger a $1 loss. Dollar figures could be assigned to each of the faces of a die. Hence we see that if each event $A_i$ has a probability, then so will the numerical values $x_i$.

The average value of $x_i$ can be approximated of course by sampling the space $N$ times, summing all the $x$'s, and dividing by $N$. As $N$ becomes bigger, this computation will give an increasingly accurate result. In terms of probabilities the formula for the expected value is

\[ \bar{x} = E(x) = \sum_{i=1}^{n} p(A_i) x_i. \]

The equivalence of this expected value with the numerical average is seen as follows: if the space is sampled $N$ times, and the number of results $[A_i, x_i]$ is $k_i$, then $ p(A_i) \simeq k_i / N $.

Superposition is an important property of the expectation operator:

\[ E(x+y) = E(x) + E(y). \]

The mean of a function of $x$ is defined using probabilities of the random variable $x$:

\[ E[f(x(\xi))] = \sum_{i=1}^{n} f(x_i) p_i . \]

Another important property of a random variable is the variance - a measure of how much the $x$ varies from its own mean:

\begin{align} \sigma ^2 &= E \left[ (x - \bar{x}) ^2 \right] \\[4pt] &= E(x^2) - \bar{x} ^2. \end{align}

The second line is apparent because $E(−2x \bar{x}) = −2 \bar{x}^2$. Note we use the symbol $ \sigma ^2 $ for variance; the standard deviation $\sigma$ is just the square root, and has the same units as does the random variable $x$.