**Part I: Probability**

- Linearity of Expectation can be summarized as E(aX+bY) = aE(X) + bE(Y). Prove this using the facts we have learned so far.
- Prove the Tower Rule, E(E(X|Y)) = E(X)
- Variance and Covariance
- Show that E(XY)=E(X)E(Y) is not a validity.
- When does E(XY)=E(X)E(Y)?
- The difference, E(XY)-E(X)E(Y) is called the
*covariance*. Why is this an apt name? - If X=y, the value is called the
*variance*. The square root of the variance is called the*standard deviation*.

**Part II: Bayesian Network**

- Model the following events as a Bayesian Network
- Hearing a police cruiser drive by, which could be caused by a police getting to a scene of a crime in progress
- That in turn could be because the officer was directly dispatched to the location, or because s/he heard something on the police radio
- A police could be directly dispatched, either because someone called 911, or another officer witnessed a crime
- Either calling 911, or having a police dispatched to a scene, may require a police report to be filled out.
- If the message was broadcast on the radio, there may be a major disturbance.

- Come up with conditional probability tables with made up (but reasonable) values for your events
- Ask three reasonably complex questions about the data
- What kind of reasoning is this?
- Answer it probabilistically.

**Part III: Projects**

Work on your projects!