Conditional Expectation

ABSTRACT

Conditional expectation is the expected value of a random variable given that a specific event has occurred. It allows for the calculation of average outcomes within a restricted subset of the sample space.

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Formula and Definition

The conditional expectation of a random variable $X$ given an event $A$ is defined as the weighted average of the values of $X$ for all outcomes in $A$:

$$E[X|A]=\frac{1}{P(A)}\sum _{a\in A}P(a)X(a)$$

This formula scales the probabilities of the outcomes within $A$ so that they sum to $1$ relative to the event $A$ itself.

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Law of Total Expectation

The Law of Total Expectation (also known as the Law of Iterated Expectations) allows you to calculate the global expectation of a random variable by partitioning the sample space into disjoint cases.

For any event $B$ and its complement $\overline{B}$:

$$E(X)=E(X|B)P(B)+E(X|\overline{B})P(\overline{B})$$

This is a powerful tool for solving problems where the outcome depends on an initial random condition or “stage”.

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Examples

1. Dice Sum with a Constraint

Calculate the expected sum $X$ of two fair dice, given that the first die is greater than 4 (Event $A$).

• Step 1: Identify $P(A)$. There are 12 outcomes where the first die is 5 or 6, so $P(A)=12/36$.
• Step 2: Sum the values of $X$ for all outcomes in $A$.
• Step 3: Apply the formula: $$\begin{array}{rl}E(X|A)& =\frac{1}{12/36}\left[\sum _{a\in A}\frac{1}{36}X(a)\right]\\ & =\frac{1}{12}(6+7+8+9+10+11+7+8+9+10+11+12)\\ & =\frac{108}{12}=\boxed{9}\end{array}$$

2. Manufacturing Quality Control

A lightbulb has a $70\%$ chance of coming from Factory A ($E[X|A]=5000$ hours) and a $30\%$ chance of coming from Factory B ($E[X|B]=6000$ hours).

Using the Law of Total Expectation:

$$\begin{array}{rl}E(X)& =E(X|A)P(A)+E(X|B)P(B)\\ & =(5000\cdot 0.7)+(6000\cdot 0.3)\\ & =3500+1800=\boxed{5300\text{ hours}}\end{array}$$

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Related Notes

• Case Analysis: The practical application of the Law of Total Expectation to break down complex probability problems.
• Expected Value: The foundational concept of the “long-run average”.
• Conditional Probabilities: The underlying rules governing how probabilities change given new information.