Definition
Concept
Binomial distribution: models number of successes in n independent Bernoulli trials. Each trial: two outcomes (success/failure). Probability of success: constant p. Trials independent.
Formal Statement
Random variable X ~ Binomial(n, p). X counts successes in n trials.
Conditions
Fixed number of trials, identical success probability, independence, binary outcome.
Properties
Discrete Distribution
Defined on integer values k = 0, 1,..., n. Probability mass assigned to each k.
Support
Support: {0, 1, 2, ..., n}. Probability outside support is zero.
Symmetry
Symmetric if p = 0.5; otherwise skewed left (p > 0.5) or right (p < 0.5).
Probability Mass Function (PMF)
Formula
PMF gives P(X = k) for k successes:
P(X = k) = C(n, k) p^k (1-p)^{n-k}Binomial Coefficient
C(n, k) = n! / (k! (n-k)!) number of ways to choose k successes.
Interpretation
Probability of exactly k successes in n trials, each success independent with probability p.
Parameters
Number of Trials (n)
Integer ≥ 1. Defines number of independent experiments.
Probability of Success (p)
Real number in [0,1]. Probability of success in each trial.
Parameter Space
Parameter vector: θ = (n, p). Fixed n, p constant across trials.
Mean and Variance
Expected Value
Mean (μ) = np. Average number of successes.
Variance
Variance (σ²) = np(1-p). Measures spread around mean.
Standard Deviation
σ = sqrt(np(1-p)).
| Parameter | Formula |
|---|---|
| Mean (μ) | np |
| Variance (σ²) | np(1-p) |
Cumulative Distribution Function (CDF)
Definition
CDF F(k) = P(X ≤ k) = Σ_{i=0}^k P(X = i).
Calculation
Summation of PMF values from 0 up to k.
Use Cases
Probability at most k successes, hypothesis testing, confidence intervals.
Moment Generating Function (MGF)
Formula
M_X(t) = E[e^{tX}] = (1 - p + p e^t)^nUsage
Derive moments (mean, variance) by differentiation.
Properties
MGF uniquely characterizes distribution.
Applications
Quality Control
Defect count in batch inspection, pass/fail testing.
Genetics
Inheritance pattern probabilities, allele frequencies.
Survey Sampling
Number of positive responses in fixed sample.
Reliability Engineering
Component failure count in system tests.
Examples
Coin Toss
Number of heads in 10 tosses of a fair coin: n=10, p=0.5.
Manufacturing Defects
Defects in 100 products, defect rate 2% (p=0.02).
Exam Passes
Number passing in class of 50 with pass probability 0.7.
| Scenario | n | p | Interpretation |
|---|---|---|---|
| Coin toss heads | 10 | 0.5 | Number of heads |
| Manufacturing defects | 100 | 0.02 | Defective items count |
| Exam passes | 50 | 0.7 | Students passing exam |
Approximations
Normal Approximation
For large n, np and n(1-p) ≥ 5, Binomial approximated by Normal(μ=np, σ²=np(1-p)). Continuity correction recommended.
Poisson Approximation
For large n, small p, with np = λ fixed, approximates Binomial by Poisson(λ).
Use Cases
Simplifies calculations, enables use of continuous distribution tools.
Parameter Estimation
Maximum Likelihood Estimation (MLE)
Given observed k successes in n trials, MLE for p is k/n.
Confidence Intervals
Use normal approximation or exact Clopper-Pearson intervals for p.
Bayesian Estimation
Beta prior conjugate; posterior is Beta(α + k, β + n - k).
Limitations
Independence Assumption
Trials must be independent; dependence invalidates model.
Constant Probability
Probability p must not vary between trials.
Binary Outcomes Only
Only two possible outcomes per trial allowed.
Fixed Number of Trials
n must be predetermined and fixed.
References
- Feller, W. "An Introduction to Probability Theory and Its Applications," Vol. 1, Wiley, 1968, pp. 152-160.
- Casella, G., Berger, R.L. "Statistical Inference," 2nd ed., Duxbury, 2002, pp. 237-244.
- Ross, S.M. "Introduction to Probability Models," 11th ed., Academic Press, 2014, pp. 58-65.
- Meyer, P.L. "Introductory Probability and Statistical Applications," Addison-Wesley, 1970, pp. 120-125.
- Johnson, N.L., Kemp, A.W., Kotz, S. "Univariate Discrete Distributions," 3rd ed., Wiley, 2005, pp. 46-60.