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Binomial Probability Distribution

It follows from Equations (2.16) and (2.20) that the probability of obtaining $ n_1$ occurrences of the outcome 1 in $ N$ statistically independent observations of a two-state system is

$\displaystyle P_N(n_1) = \frac{N!}{n_1 ! (N-n_1)!}  p^{ n_1} q^{ N-n_1}.$ (2.21)

This probability function is called the binomial probability distribution. The reason for this name becomes obvious if we tabulate the probabilities for the first few possible values of $ N$ , as is done in Table 2.1. Of course, we immediately recognize the expressions appearing in the first four rows of this table: they appear in the standard algebraic expansions of $ (p+q)$ , $ (p+q)^{ 2}$ , $ (p+q)^{ 3}$ , and $ (p+q)^{ 4}$ , respectively. In algebra, the expansion of $ (p+q)^{ N}$ is called the binomial expansion (hence, the name given to the probability distribution function), and is written

$\displaystyle (p+q)^{ N} \equiv \sum_{n=0,N}\frac{N!}{n! (N-n)!} p^{ n}  q^{ N-n}.$ (2.22)

Equations (2.21) and (2.22) can be used to establish the normalization condition for the binomial distribution function:

$\displaystyle \sum_{n_1=0,N} P_N(n_1) =\sum_{n_1=0,N} \frac{N!}{n_1! (N-n_1)!} p^{ n_1}  q^{ N-n_1}\equiv (p+q)^{ N} = 1,$ (2.23)

because $ p+q=1$ . [See Equation (2.11).]


Table 2.1: The binomial probability distribution, $ P_N(n_1)$ .
        $ n_1$    
    0 1 2 3 4
  1 $ q$ $ p$      
$ N$ 2 $ q^{ 2}$ $ 2 p q$ $ p^{ 2}$    
  3 $ q^{ 3}$ $ 3 p q^{ 2}$ $ 3 p^{ 2} q$ $ p^{ 3}$  
  4 $ q^{ 4}$ $ 4 p q^{ 3}$ $ 6 p^{ 2} q^{ 2}$ $ 4 p^{ 3} q$ $ p^{ 4}$



next up previous
Next: Mean, Variance, and Standard Up: Probability Theory Previous: Combinatorial Analysis
Richard Fitzpatrick 2016-01-25