Thm 3: For a set of n elements, take n1, n2, n3, . nr such that n1 + n2 + n3 + . nr = n. It is desired to divide (or partition) the set into r subsets, the first of which has n1 elements, the second has n2 elements, ., and the rth has nr elements. Th

WAYS TO CHOOSE SAMPLES: For a group of size n, there are: n! ways to arrange them all without replacement nr ways to choose r of them with replacement For indistinguishable subgroups n1, n2, n3. nr, there are n!/(n1 !n2!n3!.nr!) ways to arrange them

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Conditional Probability and Independence
Chapter 3, Sections 3.1 - 3.5
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Example: Suppose we toss a fair dice twice.
What is the probability that the sum of the 2 dice is 8?
Sample s

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Fall 2010
145
The Bayes Formula
For events E
S and F S such that P (E ) > 0, P (F |E ) = P (E |F )P (F ) P (E |F )P (F ) + P (E |F c )P (F c )
Proof: Using the denition of conditional probability:
P (F |E ) =
P (E |F )P (F ) P (EF

BTRY 4080 / STSCI 4080
Fall 2010
145
The Bayes Formula
For events E
S and F S such that P (E ) > 0, P (F |E ) = P (E |F )P (F ) P (E |F )P (F ) + P (E |F c )P (F c )
Proof: Using the denition of conditional probability:
P (F |E ) =
P (E |F )P (F ) P (EF

BTRY/STSCI 4080 Homework # 1 Due Date: 9/9/10 Problem numbers beginning with P (e.g., P1) denote problems from the PROBLEMS section; those beginning with T (e.g., T1) denote problems taken from the THEORETICAL EXERCISES section. 1. Ross: page 50, P1 2. Ro

BTRY/STSCI 4080 Homework # 2 Due Date: 9/16/10 Problem numbers beginning with P (e.g., P1) denote problems from the PROBLEMS section; those beginning with T (e.g., T1) denote problems taken from the THEORETICAL EXERCISES section. 1. Ross: p 51, P11 2. Ros

BTRY/STSCI 4080 Homework # 3 Due Date: 9/23/10 Problem numbers beginning with P (e.g., P1) denote problems from the PROBLEMS section; those beginning with T (e.g., T1) denote problems taken from the THEORETICAL EXERCISES section. 1. Ross: p 16, P8 2. Ross

BTRY/STSCI 4080 Homework # 4 Due Date: 9/30/10 Problem numbers beginning with P (e.g., P1) denote problems from the PROBLEMS section; those beginning with T (e.g., T1) denote problems taken from the THEORETICAL EXERCISES section. 1. Ross: p 51, P16 2. Ros

BTRY/STSCI 4080 Homework # 5 Due Date: 10/7/10 Problem numbers beginning with P (e.g., P1) denote problems from the PROBLEMS section; those beginning with T (e.g., T1) denote problems taken from the THEORETICAL EXERCISES section. 1. Ross: p 102, P15 2. Ro

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Fall 2010
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Set Theory: Chapter 2.2
What is set theory?
Notation and rules that permit manipulation of sets in a logically coherent way. A branch of mathematics with several parallels to, and differences from,
algebra and calculus

BTRY 4080 / STSCI 4080
Fall 2010
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A Simple Set Function: Indicators
Set theory: tools useful for mathematically manipulating sets. Important
because calculating the probability of an event involves representing it as a set and then manipulating it in w

BTRY 4080 / STSCI 4080
Fall 2010
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Permutations and Combinations, Sections 1.3-1.5
Def: For an integer n
1, we dene n factorial as follows: n! := n (n 1) 2 1.
By convention:
0! = 1.
Given n distinguishable objects, n! is the number of unique ways in wh

CHAPTER 1: Basic Principle of Counting r experiments to be performed, each with n possible outcomes, total of n 1 x n2 x.x nr total outcomes. General formula for permutations (order matters so abc bca: Given n objects, there are n! different ways t

WAYS TO CHOOSE SAMPLES: For a group of size n, there are: n! ways to arrange them all without replacement nr ways to choose r of them with replacement For indistinguishable subgroups n1, n2, n3. nr, there are n!/(n1 !n2!n3!.nr!) ways to arrange them

Chapter 2
Conditional Probability
Suppose that we have partial knowledge of the result of a trial of the experiment - we know that event A occurred on the trial - but we do not know which outcome in A occurred; only that the

Chapter 3
Independence
Suppose that it is known that event A has occurred. We update the probability of the event B P(BA) from P(B) to P(B|A) = P(A) P(B|A) can be larger, smaller, or the same as P(B) If P(B|A) = P(B),

Chapter 4
Random Variables
Random variables are used to model phenomena in which the experimental outcomes are numbers, e.g. 1, 2, 3, or 3.213678. instead of labels such as Head or Tail or Luke or Darth Example: = , or = Z We do not kn

Chapter 5
Many Random Variables
A random variable models phenomena in which the experimental outcomes are numbers A random variable measures one physical parameter Different random variables measure different parameters Example: Req

Chapter 6
Limit Theorems
In this chapter, we consider various bounds on probabilities that don't depend on specific forms of CDFs/pdfs/pmfs Relative frequencies and large numbers of trials Laws of large numbers Asymptotic distributions o

Example. Suppose there are 8 men and 8 women. How many ways can we choose a committee that has 2 men and 2 women? 8 8 Answer. We can choose 2 men in ways and 2 women in ways. The number of committees is then 2 2 8 8 the product: . 2 2 Let us look at

1. Combinatorics. The first basic principle is to multiply. Suppose we have 4 shirts of 4 different colors and 3 pants of different colors. How many possibilities are there? For each shirt there are 3 possibilities, so altogether there are 4 3 = 12

Example. Suppose 10 people put a key into a hat and then withdraw one randomly. What is the probability at least one person gets his/her own key? Answer. If Ei is the event that the ith person gets his/her own key, we want P(10 Ei ). One can show, i=

Example. In n trials, let X be the number of successes. A discrete random variable is one that can only take countably many values. For a discrete random variable, we define the probability mass function or the density by p(x) = P(X = x). Here P(X =

(x), the area under the standard Gaussian density function to the left of x
x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 0.00 0.5000 0.5398 0.5793 0.6179

2. The probability set-up. We will have a sample space, denoted S (sometimes ) that consists of all possible outcomes. For example, if we roll two dice, the sample space would be all possible pairs made up of the numbers one through six. An event is

Suppose P(E | F ) = P(E), i.e., knowing F doesn't help in predicting E. Then E and F are independent. What we have said is that in this case P(E | F ) = P(E F ) = P(E), P(F )
or P(E F ) = P(E)P(F ). We use the latter equation as a definition: We s