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Probability and Statistics for Finance by Svetlozar T. Rachev, Markus Höechstöetter, Sergio M. Focardi, Frank J. Fabozzi CFA

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Appendix C. Binomial and Multinomial Coefficients

In this appendix, we explain the concept of the binomial and multinomial coefficients used in discrete probability distributions described in Chapter 9.

Binomial Coefficient

The binomial coefficient is defined as

Equation C.1. 

Binomial Coefficient

for some nonnegative integers k and n with 0 ≤ kn. For the binomial coefficient, we use the factorial operator denoted by the "!" symbol. As explained in Appendix A, a factorial is defined in the set of natural numbers N that is k = 1, 2, 3, ... as

Equation C.2. 

Binomial Coefficient

For k = 0, we define 0! ≡ 1.

Derivation of the Binomial Coefficient

In the context of the binomial distribution described in Chapter 9, we form the sum X of n independent and identically distributed Bernoulli random variables Yi with parameter p or, formally,

Derivation of the Binomial Coefficient

Special Case n = 3

We illustrate the special case where n = 3 using a B(3,0.4) random variable X; that is, X is the sum of three independent B(0.4) distributed random variables Y1, Y2, and Y3. All possible values for X are contained in the state space Ω={0, 1, 2, 3}. As we will see, some of these k ∈ Ω can be obtained in different ways.

We start with k = 0. This value can only be obtained when all Yi are ...

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