N: Statistical distributions flashcards

Topic 15: N: Statistical distributions

Rule for a discrete probability distribution

Every probability is between 0 and 1, and Σ P(X = x) = 1.

Discrete uniform distribution

Every value is equally likely, for example the score on a fair dice (each 1/6).

Conditions for a binomial distribution B(n, p)

A fixed number n of independent trials, each with two outcomes and the same probability of success p. X is the number of successes.

Binomial: P(X = r), mean and variance

P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ. Mean np; variance np(1 − p).

X ~ B(10, 0.5). Find P(X = 5).

¹⁰C₅ × 0.5¹⁰ = 252/1024 = 0.246 (3 s.f.).

Key features of the normal distribution N(μ, σ²)

Continuous, bell-shaped and symmetrical about the mean μ, with variance σ². Points of inflection at x = μ ± σ.

Standardising a normal variable

Z = (X − μ)/σ, where Z ~ N(0, 1) has mean 0 and standard deviation 1.

When can B(n, p) be approximated by a normal distribution, and how?

For large n with p not near 0 or 1. Use N(np, np(1 − p)) with a continuity correction: P(X ≥ 55) becomes P(Y > 54.5).

Binomial or normal model?

Binomial: a discrete count of successes in a fixed number of trials. Normal: a continuous measurement (mass, time, length) that is roughly symmetrical and bell-shaped.