Tutorial 8, STAT1301 Fall 2010, 16NOV2010, By Joseph Dong

Recall: A Partition on a Set 2

Recall: Conditioning on a Partition Shares the same idea with Divide and Conquer Casewise enumeration A Tree-diagram 3

Recall: Conditioning on a Partition (contd) 4

Recall: What does an R.V. do to its State Space? An r.v. cuts the state space into blocks. On each of these blocks, the r.v. sends all points there to a common atom in the sample space. An r.v. causes a partition on the state space. Conversely, given a partition on the state space, you can also define random variables on it so that it conforms the partition by taking one value for each block. Random Variable 5

Conditioning an Event on an R.V. 6

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Conditional, Marginal, and Joint densities 10

11 Handout Problem 1

Recall: Whats the Expectation of a random variable 12

For examples, 13

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Handout Problem 2 15 Warm-up exercise

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18 Degeneration preserves overall center of mass

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20 Conditional Variance

The key observations are Obs1: To find the center of mass of a piece of material, you can divide it into a few blocks, find their centers of mass, and then find the center of mass of these block centers of masses. The initial division of the piece is quite arbitrary. This fundamental law of physics supports the many nice properties of expectation in the calculus of probability. Obs2: A random variable partitions its state space into a collection of atom-valued blocks. This suggests using random variable as a general device to divide the piece mentioned in Obs1. Such a random variable is called the conditioner. 21 Summary: Conditional Expectation

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23 Choosing Conditioner

Handout problem 3 Handout problem 4 Handout problem 5 Handout problem 6 24 Exercises