Question 1

What is the binomial distribution?

Accepted Answer

The binomial distribution describes the number of successes k in n independent Bernoulli trials, each with probability p of success. Four conditions must hold: fixed number of trials, each trial has only two outcomes (success/failure), trials are independent, and p is constant across trials. Examples: number of heads in 10 coin flips; defective units in a 100-item batch; patients responding to a drug in a trial.

Question 2

What is the difference between P(X=k), P(X≤k), and P(X≥k)?

Accepted Answer

P(X=k) is the exact probability of getting exactly k successes (the PMF). P(X≤k) is the cumulative probability of getting k or fewer successes — the CDF. P(X≥k) is the probability of getting k or more successes = 1 − P(X≤k−1). Example (n=10, p=0.3): P(X=3) ≈ 0.267, P(X≤3) ≈ 0.650, P(X≥3) ≈ 0.617.

Question 3

What are the mean and variance of a binomial distribution?

Accepted Answer

Mean (expected value): μ = n × p. Variance: σ² = n × p × (1 − p). Standard deviation: σ = √(n × p × (1 − p)). Example: n=100 trials, p=0.3 → mean = 30, variance = 21, SD ≈ 4.58. You expect about 30 successes on average with typical variation of ±4.6.

Question 4

When can I approximate the binomial with the normal distribution?

Accepted Answer

The normal approximation is reasonable when both n×p ≥ 5 and n×(1−p) ≥ 5. Under these conditions, X ≈ Normal(μ=np, σ²=np(1−p)). Apply a continuity correction: P(X=k) ≈ P(k−0.5 < Y < k+0.5) where Y is normal. For small p (p < 0.05) and np < 10, the Poisson approximation (λ=np) is more accurate.

Question 5

How is the binomial distribution used in A/B testing?

Accepted Answer

In A/B testing, conversions follow a binomial distribution: n = visitors, p = conversion rate, k = conversions observed. To test if a new design differs from the control (known p₀), compute P(X ≥ k_obs | n, p₀) for the one-tailed p-value. For large n, a one-proportion z-test is equivalent. Example: control rate 5%, new variant gets 12 conversions from 150 visitors — compute P(X ≥ 12 | n=150, p=0.05) ≈ 0.012, significant at α=0.05.

Question 6

What is the difference between binomial and Poisson distributions?

Accepted Answer

Binomial: fixed number of trials n, each with probability p. Use when n is moderate and p is not extremely small. Poisson: models rare events with no fixed upper count; λ = average rate. Use when n is very large and p very small (np is moderate). Rule: if n > 20 and p < 0.05 (np < 10), use Poisson with λ = np. The Poisson is the limit of binomial as n→∞, p→0 with λ = np constant.

Scenario	Use	Conditions	Why
Fixed n trials, known p, counting successes	Binomial (exact)	Any n, any p — always valid	Exact model for the problem
Very large n (>1000), small p (<0.05), np < 10	Poisson (λ=np)	n→∞, p→0, np finite	Computationally simpler, essentially exact
Large n, np ≥ 5 and n(1−p) ≥ 5	Normal approx	n×p ≥ 5 and n×(1−p) ≥ 5	Central limit theorem applies
Small n (< 20) or p near 0 or 1	Binomial (exact)	Skewed distribution	Normal approx fails at extremes
No upper bound on count, counting in interval	Poisson	Continuous interval, rare events	Binomial requires fixed n
Proportion CI or z-test for conversion rate	Normal (CLT)	n large, p not extreme	Allows use of z-table / confidence intervals

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