chapter 4 - binomial coefficients
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Chapter 4Binomial Coefficients
4.1 Binomial Coefficient Identities4.2 Binomial Inversion Operation4.3 Applications to Statistics4.4 The Catalan Recurrence
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2 Chapter 4 Binomial Coefficients
4.1 BINOMIAL COEFF IDENTITIES
Table 4.1.1
Section 4.1 Binomial Coeff Identities 3
4 Chapter 4 Binomial Coefficients
Combinatorial vs. Algebraic Proofs
Symmetry
Section 4.1 Binomial Coeff Identities 5
Row-Sum Property
6 Chapter 4 Binomial Coefficients
Column-Sum Property
Section 4.1 Binomial Coeff Identities 7
8 Chapter 4 Binomial Coefficients
Diagonal-Sum Properties
Section 4.1 Binomial Coeff Identities 9
10 Chapter 4 Binomial Coefficients
Section 4.1 Binomial Coeff Identities 11
12 Chapter 4 Binomial Coefficients
Products of Binomial Coefficients
Section 4.1 Binomial Coeff Identities 13
14 Chapter 4 Binomial Coefficients
Section 4.1 Binomial Coeff Identities 15
16 Chapter 4 Binomial Coefficients
Vandermonde Convolution
Section 4.1 Binomial Coeff Identities 17
Summary of Binomial Coeff Identities
Table 4.1.2
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Parity of Binomial Coefficients
Section 4.1 Binomial Coeff Identities 19
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Section 4.1 Binomial Coeff Identities 21
22 Chapter 4 Binomial Coefficients
Section 4.1 Binomial Coeff Identities 23
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Section 4.2 Binomial Inversion 25
4.2 BINOMIAL INVERSION
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Section 4.2 Binomial Inversion 27
Some Basic Examples of Inversions
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Section 4.2 Binomial Inversion 29
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Section 4.2 Binomial Inversion 31
Derangements
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Section 4.2 Binomial Inversion 33
More Examples of Inversions
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Section 4.2 Binomial Inversion 35
36 Chapter 4 Binomial Coefficients
4.3 APPLICATIONS TO STATISTICS
Probability and Random Variables
Section 4.3 Applications to Statistics 37
Mean and Variance
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Section 4.3 Applications to Statistics 39
Binomial Distribution
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Section 4.3 Applications to Statistics 41
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Section 4.3 Applications to Statistics 43
Unbiased Estimator of the Mean
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Section 4.3 Applications to Statistics 45
Unbiased Estimator of the Variance
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Section 4.3 Applications to Statistics 47
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4.4 THE CATALAN RECURRENCE
Binary Trees
Section 4.4 The Catalan Recurrence 49
Fig 4.4.1
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Fig 4.4.2
Section 4.4 The Catalan Recurrence 51
Nested Parentheses
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Subdiagonal Paths
Fig 4.4.3
Section 4.4 The Catalan Recurrence 53
Solving the Catalan Recurrence
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Section 4.4 The Catalan Recurrence 55
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Generalized Binomial Theorem
Section 4.4 The Catalan Recurrence 57
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Alternative Proof of the Catalan Formula
Section 4.4 The Catalan Recurrence 59
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Section 4.4 The Catalan Recurrence 61
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