Definition
Binomial Distribution
Discrete probability distribution describing number of successes in n independent Bernoulli trials each with success probability p.
Bernoulli Trial
Random experiment with exactly two outcomes: success (probability p) or failure (probability 1-p).
Random Variable
Let X = number of successes in n trials. X ~ Binomial(n, p).
Assumptions
Fixed Number of Trials
Number of trials n is predetermined and finite.
Independence
Trials are statistically independent; outcome of one does not affect another.
Constant Probability
Success probability p remains constant across all trials.
Probability Mass Function
PMF Formula
Probability X equals k successes in n trials:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)Combination Term
C(n, k) = n! / [k! (n - k)!], counts ways to choose k successes.
Support
k ∈ {0, 1, 2, ..., n} integer values only.
Parameters
Number of Trials (n)
Positive integer, total count of Bernoulli trials.
Success Probability (p)
Real number in [0,1], probability of success per trial.
Parameter Space
n ∈ ℕ, p ∈ [0,1].
Mean and Variance
Expected Value
Mean number of successes: E[X] = np.
Variance
Spread of distribution: Var(X) = np(1 - p).
Standard Deviation
σ = sqrt(np(1 - p)).
| Parameter | Formula |
|---|---|
| Mean (E[X]) | np |
| Variance (Var(X)) | np(1-p) |
Cumulative Distribution Function
Definition
CDF F(k) = P(X ≤ k) = sum_{i=0}^k P(X = i).
Computation
Sum of PMF values for i = 0 to k inclusive.
Properties
Non-decreasing, right-continuous, F(n) = 1.
Moment Generating Function
MGF Formula
M_X(t) = E[e^{tX}] = (1 - p + p e^{t})^nUse
Derives moments: mean, variance, higher moments.
Derivatives
First derivative at t=0 gives E[X], second derivative gives E[X^2].
Applications
Quality Control
Modeling defective items in production sample.
Medical Studies
Number of patients responding to treatment.
Finance
Modeling success/failure of investments.
Reliability Engineering
Failures in system components tested independently.
Examples
Coin Toss
Number of heads in 10 tosses of a fair coin: Binomial(10, 0.5).
Exam Questions
Number of correct answers in multiple choice test with guessing.
Drug Effectiveness
Number of patients cured out of fixed sample.
| Scenario | Parameters (n, p) | Interpretation |
|---|---|---|
| Coin Toss | n=10, p=0.5 | Count heads in 10 tosses |
| Multiple Choice Test | n=20, p=0.25 | Correct guesses out of 20 |
| Drug Trial | n=50, p=0.6 | Patients cured in trial |
Approximations
Normal Approximation
For large n, Binomial approximated by Normal(np, np(1-p)) with continuity correction.
Poisson Approximation
If n large, p small, λ = np fixed, approximate Binomial by Poisson(λ).
Conditions
Normal valid if np ≥ 5 and n(1-p) ≥ 5. Poisson valid if np ≤ 10 and p ≤ 0.1.
Normal approx: P(X ≤ k) ≈ Φ((k + 0.5 - np) / sqrt(np(1-p)))Statistical Inference
Parameter Estimation
Estimate p by sample proportion p̂ = X/n, unbiased and consistent.
Confidence Intervals
Exact (Clopper-Pearson) and approximate (Wald, Wilson) intervals for p.
Hypothesis Testing
Test null hypothesis about p using binomial test or normal approximation.
Limitations
Trial Independence
Assumes independent trials, violated in correlated data.
Constant Probability
p must be constant; unsuitable for changing probabilities over trials.
Discrete Outcomes Only
Only models count of successes, no partial or continuous outcomes.
References
- Feller, W., "An Introduction to Probability Theory and Its Applications," Vol. 1, Wiley, 1968, pp. 150-180.
- Ross, S.M., "Introduction to Probability Models," 11th Ed., Academic Press, 2014, pp. 90-105.
- Casella, G., Berger, R.L., "Statistical Inference," 2nd Ed., Duxbury, 2002, pp. 300-320.
- Lehmann, E.L., Romano, J.P., "Testing Statistical Hypotheses," 3rd Ed., Springer, 2005, pp. 120-130.
- Johnson, N.L., Kemp, A.W., Kotz, S., "Univariate Discrete Distributions," 3rd Ed., Wiley, 2005, pp. 75-110.