Probability Theory

## Objectives

This assignment is designed to:

• provide practice with some essential probability theory
• help you become more fluent with the terminology and the techniques

## Instructions

This is a mathematical homework assignment. Show your work. Be clear and concise. Type your assignment.<br>

I strongly recommend you work through these exercises as soon as possible for their instructional content. If you have general questions about the assignment, please post on the Google Group. Finish early, and earn the early bonus.

## Exercises

### Question 1: Useful theorems in probability theory

[50 points; 10 per sub-problem]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.1)

Use the Set Theory Identities and Axioms of Probability Theory to prove each of the following five statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply good proof technique: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom.

1. $P(B - A) = P(B) - P(A \cap B)$
2. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule)
• Hint: refer to part result #1 as a part of your proof of part #2
3. $P(\varnothing) = 0$
4. $P(\neg A) = 1 - P(A)$
• Hint: refer to part result #1 as a part of your proof of part #4
5. $A \subseteq B \Rightarrow P(A) \leq P(B)$
• Hint: let $C = B - A$ (set difference).

### Question 2: Joint Probability

[10 points]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.3)

Compute the probability of the event 'A period occurs after a three-letter word, and this period indicates an abbreviation (not an end-of-sentence marker)'.

• Let is-abbreviation denote the event “this period indicates an abbreviation”, and let three-letter-word denote “a period occurs after a three letter word”
• Assume the following probabilities:
• $P($is-abbreviation$|$three-letter-word$) = 0.8$
• $P($three-letter-word$) = 0.0003$

### Question 3: Independence

[10 points]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.4)

Are $X$ and $Y$ as defined in the following table independently distributed?

<table border=1 cellspacing=0> <tr>

 $x$
0
0
1
1
</tr> <tr>
 $y$
0
1
0
1
</tr> <tr>
 $P(X=x, Y=y)$
0.32
0.08
0.48
0.12
</tr> </table>

### Question 4: Factoring Joint Distributions

[10 points]

1. How many possible ways can you completely factor the joint distribution $P(X_1, X_2, X_3, X_4, X_5, X_6)$?
2. For some arbitrary joint probability distribution on six random variables $P(X_1, X_2, X_3, X_4, X_5, X_6)$, apply the chain rule to completely factor this distribution in one way.

### Question 5: Conditional Probability

[10 points]

Apply the definition of conditional probability to prove Bayes’ Law. Use the same standard of proof as in problem #1 above.

### Question 6: Chain rule

[10 points]

Prove the chain rule, namely that $P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot \ldots\cdot P(A_n | \cap^{n-1}_{i=1} A_i)$. Justify each step. Use the same standard of proof as in problem #1 above.

• Hint: you could use induction (but that isn't the only way).

## Submission

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