Continuous and discrete distributions
Continuous and discrete distributions
This domain tests probability models: which distribution applies to a given random variable, the parameters that define it, and how to compute probabilities, means, and variances. Expect binomial, Poisson, uniform, normal, and exponential items, plus normal approximations and continuity corrections.
Discrete models
- Binomial: fixed n independent trials, two outcomes, constant p. P(X = k) = C(n,k) p^k (1−p)^(n−k); μ = np, σ² = np(1−p).
- Poisson: counts of rare events in a fixed interval with mean λ. P(X = k) = e^−λ λ^k / k!; μ = σ² = λ.
- Geometric: trials until the first success; P(X = k) = (1−p)^(k−1)p; μ = 1/p.
- Hypergeometric: sampling without replacement from a finite population; trials are not independent.
Continuous models
- Probability is area under the density curve; P(X = c) = 0, and a density may exceed 1.
- Uniform on [a, b]: density 1/(b−a), μ = (a+b)/2, σ² = (b−a)²/12.
- Normal: standardize with z = (x − μ)/σ; use the empirical rule (68–95–99.7) as a check.
- Exponential: waiting time between Poisson events, P(X > t) = e^−λt, memoryless.
Worked example
A call center receives a mean of 3 calls per hour, Poisson. P(X = 0) = e^−3 · 3^0/0! = e^−3 ≈ 0.0498. P(X ≥ 1) = 1 − 0.0498 = 0.9502. Because the Poisson variance equals λ = 3, the standard deviation is √3 ≈ 1.73, not 3.
Traps
- Applying binomial formulas when sampling is without replacement from a small population; use hypergeometric.
- Omitting the continuity correction when approximating a discrete count with the normal curve (use 2.5 for P(X ≤ 2)).
- Using the normal approximation when np or n(1−p) is below about 10.
- Confusing λ in the Poisson (mean count) with λ in the exponential (rate per unit time).
- Treating density height as probability.