# Lab 11: Bayesian Networks

Part I: Probability

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

Part II: Bayesian Network

1. Model the following events as a Bayesian Network
1. Hearing a police cruiser drive by, which could be caused by a police getting to a scene of a crime in progress
2. That in turn could be because the officer was directly dispatched to the location, or because s/he heard something on the police radio
3. A police could be directly dispatched, either because someone called 911, or another officer witnessed a crime
4. Either calling 911, or having a police dispatched to a scene, may require a police report to be filled out.
5. If the message was broadcast on the radio, there may be a major disturbance.
2. Come up with conditional probability tables with made up (but reasonable) values for your events
3. Ask three reasonably complex questions about the data
1. What kind of reasoning is this?
2. Answer it probabilistically.

Part III: Projects

Work on your projects!