Statistical Distributions

What is a discrete probability distribution?

A list (or formula) giving each possible value of a discrete random variable along with its probability. The probabilities must sum to 1.

Example of a discrete distribution defined by a formula

P(X = x) = 0.05x(x + 1) for x = 1, 2, 3 — check that the probabilities add to 1.

What is the binomial distribution used to model?

The number of successes in a fixed number of independent trials, each with the same probability of success.

Conditions for a binomial distribution (BINS)

Binary outcomes (success/failure); Independent trials; fixed Number of trials n; Same probability p of success in each trial.

Notation for the binomial distribution

X ~ B(n, p), where n is the number of trials and p is the probability of success.

Binomial probability formula

P(X = x) = C(n, x) p^x (1 − p)^(n − x), for x = 0, 1, …, n.

What is the difference between a condition and an assumption (binomial)?

A condition is a requirement of the model (e.g. fixed n); an assumption is a real-world claim we're making to justify the model (e.g. that the trials really are independent in this context).

How do you find binomial probabilities on a calculator?

Use the binomial PD function for P(X = x) and the binomial CD function for P(X ≤ x).

Mean of a binomial distribution X ~ B(n, p)

μ = np.

Variance of a binomial distribution X ~ B(n, p)

σ² = np(1 − p) = npq, where q = 1 − p.

What is the normal distribution used to model?

A continuous random variable whose distribution is symmetric and bell-shaped, e.g. heights, masses, measurement errors.

Notation for the normal distribution

X ~ N(μ, σ²), where μ is the mean and σ² is the variance.

What is the standard normal distribution?

Z ~ N(0, 1) — a normal distribution with mean 0 and variance 1.

Standardisation formula

Z = (X − μ) / σ, converting X ~ N(μ, σ²) to the standard normal.

About what proportion of normal data lies within μ ± σ?

Approximately two-thirds (≈68%).

About what proportion of normal data lies within μ ± 2σ?

Approximately 95%.

About what proportion of normal data lies within μ ± 3σ?

Almost all (≈99.7%).

Where do the points of inflection of a normal curve occur?

At x = μ ± σ.

How do you find P(X < x) for X normal on a calculator?

Use the normal CD function with the given mean and standard deviation.

How do you find x such that P(X < x) = p (inverse normal)?

Use the inverse normal function on the calculator with the probability, mean and standard deviation.

What links a histogram to the normal model?

When data is bell-shaped and roughly symmetric, a histogram of relative frequency can be approximated by the normal probability density curve.

When can a binomial distribution be approximated by a normal distribution?

When n is large (and np and n(1−p) are not too small). Use μ = np and σ² = np(1−p) for the approximating normal.

Are calculations using the normal approximation to the binomial examinable on OCR A H240?

No — OCR A only requires understanding the approximation and the formulas for μ and σ², not calculations with it.

How do you choose between binomial and normal models?

Binomial: counting successes in a fixed number of independent trials. Normal: a continuous variable that is approximately symmetric/bell-shaped about a mean.

When might the binomial model be inappropriate?

When trials are not independent, the probability of success varies, or the number of trials is not fixed.

When might the normal model be inappropriate?

When the data is strongly skewed, multi-modal, bounded in a way the normal cannot handle, or the variable is clearly discrete with few values.