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{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*}

--- ## 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.| --- ## Related Notes - **[[Random Variables]]**: Expected value is only defined for numerical random variables. - **[[Conditional Expectation#law-of-total-expectation|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.