Expected Value

ABSTRACT

The expected value ($E[X]$) is the long-run average value of a random variable over many repetitions of an experiment. It is the probability-weighted average of all possible outcomes.

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Formal Definition

The expectation of a random variable $X$ on a sample space $S$ can be calculated in two equivalent ways:

1. By Outcome: Summing over every individual outcome $s$ in the sample space $S$.

   $$E(X)=\sum _{s\in S}P(s)X(s)$$

2. By Value: Summing over every possible value $r$ that the random variable $X$ can take.

   $$E(X)=\sum _{r\in X(S)}P(X=r)\cdot r$$

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Examples

1. Rolling a Single Die

Let $X$ be the number showing on a fair six-sided die after one roll. Each outcome has a probability of $1/6$.

$$\begin{array}{rl}E(X)& =1\cdot P(X=1)+2\cdot P(X=2)+3\cdot P(X=3)+4\cdot P(X=4)+5\cdot P(X=5)+6\cdot P(X=6)\\ & =1\cdot \left(\frac{1}{6}\right)+2\cdot \left(\frac{1}{6}\right)+3\cdot \left(\frac{1}{6}\right)+4\cdot \left(\frac{1}{6}\right)+5\cdot \left(\frac{1}{6}\right)+6\cdot \left(\frac{1}{6}\right)\\ & =\boxed{3.5}\end{array}$$

2. Sum of Two Dice

Let $X$ be the random variable representing the sum of pips after rolling two fair dice. The possible values for $X$ range from $2$ to $12$.

To find the expectation, we multiply each possible sum by its corresponding probability:

$$\begin{array}{rl}E(X)& =2\left(\frac{1}{36}\right)+3\left(\frac{1}{18}\right)+4\left(\frac{1}{12}\right)+5\left(\frac{1}{9}\right)+6\left(\frac{5}{36}\right)+7\left(\frac{1}{6}\right)+8\left(\frac{5}{36}\right)+9\left(\frac{1}{9}\right)+10\left(\frac{1}{12}\right)+11\left(\frac{1}{18}\right)+12\left(\frac{1}{36}\right)\\ & =\boxed{7}\end{array}$$

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Properties and Applications

Concept	Application
Linearity of Expectation	A shortcut for the two-dice problem: $E[X_1+X_2]=E[X_1]+E[X_2]=3.5+3.5=7$.
Conditional Expectation	Used when calculating the average outcome given that a certain event has already occurred.
Distributions	Every standard distribution (Binomial, Geometric, etc.) has a predefined formula for its expected value.

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Related Notes

• Random Variables Definition: Expected value is only defined for numerical random variables.
• Law of Total Expectation: A method for calculating $E[X]$ by partitioning the sample space into cases.
• Case Analysis: Useful for finding the expected value of complex, multi-stage experiments.