Math 2.0

Probability theory and stochastic processes Definition

Binomial distribution

Builds on Probability mass functionBinomial coefficientBernoulli trial

Definition

Let n≥0n \ge 0 be an integer and let p∈[0,1]p \in [0, 1]. The binomial distribution with parameters nn and pp is the distribution on {0,1,…,n}\{0, 1, \dots, n\} with the probability mass function p(k)=(nk)pk(1−p)n−kfor k=0,1,…,n.p(k) = \binom{n}{k} p^k (1 - p)^{n-k} \quad \text{for } k = 0, 1, \dots, n. Here (nk)\binom{n}{k} is the binomial coefficient. One writes X∼B⁡(n,p)X \sim \operatorname{B}(n, p) or X∼Bin⁡(n,p)X \sim \operatorname{Bin}(n, p).

Remarks

The values add up to 11 by the binomial theorem: ∑k(nk)pk(1−p)n−k=(p+1−p)n=1\sum_{k} \binom{n}{k} p^k (1 - p)^{n-k} = (p + 1 - p)^n = 1. The number of successes in nn independent Bernoulli trials with success probability pp has this distribution. The factor (nk)\binom{n}{k} counts the positions of the kk successes. The mean is npnp and the variance is np(1−p)np(1 - p). The probability of exactly 22 heads in 44 tosses of a fair coin is (42)/24=6/16\binom{4}{2} / 2^4 = 6/16.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Bernoulli trial concept
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. The rationals form a field statement
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Sigma-algebra definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability space definition
  42. Random variable definition
  43. Discrete random variable definition
  44. Probability mass function definition
  45. Binomial distribution definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_distribution, Probability mass function
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MSC60E05
DefinesBinomial distribution
UsesProbability mass function, Binomial coefficient, Bernoulli trial

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