Binomial Probability Distribution Family

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A Binomial Probability Distribution Family, [math]\displaystyle{ B(n,p) }[/math], is a finite support discrete probability distribution family for a binomial random variable.



  • (Wikipedia, 2015) ⇒ Retrieved:2015-6-15.
  • (Wikipedia, 2015) ⇒ Retrieved:2015-6-15.
    • In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(np). The probability of getting exactly k successes in n trials is given by the probability mass function: : [math]\displaystyle{ f(k;n,p) = \Pr(X = k) = {n\choose k}p^k(1-p)^{n-k} }[/math] for k = 0, 1, 2, ..., n, where : [math]\displaystyle{ {n\choose k}=\frac{n!}{k!(n-k)!} }[/math] is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: we want exactly k successes (pk) and n − k failures (1 − p)n − k. However, the k successes can occur anywhere among the n trials, and there are [math]\displaystyle{ {n\choose k} }[/math] different ways of distributing k successes in a sequence of n trials.

      In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as : [math]\displaystyle{ f(k,n,p)=f(n-k,n,1-p). }[/math] Looking at the expression ƒ(knp) as a function of k, there is a k value that maximizes it. This k value can be found by calculating : [math]\displaystyle{ \frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)} }[/math] and comparing it to 1. There is always an integer M that satisfies :[math]\displaystyle{ (n+1)p-1 \leq M \lt (n+1)p. }[/math]

      ƒ(knp) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which ƒ is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable (most likely) outcome of the Bernoulli trials and is called the mode. Note that the probability of it occurring can be fairly small.

       Recurrence relation [math]\displaystyle{ \left\{p (n-k) \text{Prob}(k)+(k+1) (p-1) \text{Prob}(k+1)=0,\text{Prob}(0)=( 1-p)^n\right\} }[/math]



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    • QUOTE: The binomial distribution applies when there are two possible outcomes. You know the probability of obtaining either outcome (traditionally called "success" and "failure") and want to know the chance of obtaining a certain number of successes in a certain number of trials.
      • How many trials (or subjects) per experiment?
      • What is the probability of "success" in each trial or subject?