Conditional expectation
Conditional expectation
The conditional expectation of a discrete random variable $X$ given that the random variable $Y$ equals $y_i$ is given by: $$ \mathbb{E}(X|Y=y_i) = \sum_{j=0}^\infty x_j P(X=x_j | Y=y_i ) $$ Conditional expectation values are useful because we can use them to calculate absolute expectation values using the conditional expectation theorem: $$ \mathbb{E}(X) = \sum_{i=0}^\infty \mathbb{E}(X|Y=y_i) P(Y=y_i) $$ Another useful result that is derived using similar logic involves the fact that the expectation $\mathbb{E}(XY)$ can be calculated as: $$ \mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y) $$ If the random variables $X$ and $Y$ are independent. This equation does not hold in general when the variables are not independent.
Syllabus Aims
- You should be able to explain what a conditional expectation value of a random variable measures.
- You should be able to write out the conditional expectation theorem.
- You should be able to calculate the mean and variance of the geometric random variable using the conditional expectation theorem.
- You should be able to explain when the expectation $\mathbb{E}(XY)$ can be calculated as $\mathbb{E}(X)\mathbb{E}(Y)$.
Description and link | Module | Author | ||
| A description of the conditional expectation theorem and an explanation of how it can be used to calculate expectation values. | SOR3012 | J. F. McCann |
Description and link | Module | Author | ||
| Using the conditional expectation theorem to calculate the expectation and variance of the geometric random variable | SOR3012 | G. Tribello |
Description and link | Module | Author | ||
| Problems involving using the conditional expectation theorem to calculate expectation values. | SOR3012 | G. Tribello |
Contact Details
School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN
Email: g.tribello@qub.ac.uk
Website: mywebsite