ABSTRACT
The expected value () 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.
Formal Definition
The expectation of a random variable on a sample space can be calculated in two equivalent ways:
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By Outcome: Summing over every individual outcome in the sample space .
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By Value: Summing over every possible value that the random variable can take.
Examples
1. Rolling a Single Die
Let be the number showing on a fair six-sided die after one roll. Each outcome has a probability of .
\begin{align*} 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{align*}$$ ### 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 $12$. ![[Pasted image 20251019132329.png]] To find the expectation, we multiply each possible sum by its corresponding probability:\begin{align*}
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{align*}