Probability Distributions
Discrete probability distributions, expected value, binomial distribution, normal distribution
Discrete Probability Distributions
A discrete probability distribution lists all possible values of a discrete random variable X and their associated probabilities.
A variable whose value is determined by the outcome of a random experiment. Denoted by a capital letter (e.g. X).
- All probabilities must be between 0 and 1: 0 ≤ P(X = x) ≤ 1
- All probabilities must sum to 1: ΣP(X = x) = 1
E(X) = Σ[x × P(X = x)] The expected value is the long-run average value of the random variable.
X can take values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X).
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.
- Fixed number of trials (n)
- Each trial has only two outcomes: success or failure
- Probability of success (p) is constant for each trial
- Trials are independent
P(X = r) = ⁿCᵣ × pʳ × (1−p)ⁿ⁻ʳ Where n = number of trials, r = number of successes, p = probability of success Mean: E(X) = np Variance: Var(X) = np(1−p)
Always check the four conditions before using the binomial distribution. The notation is X ~ B(n, p).
Normal Distribution
The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. Many natural measurements (height, weight, IQ) follow a normal distribution.
- Symmetric about the mean
- Mean = Median = Mode
- Bell-shaped curve
- Total area under the curve = 1
- Approximately 68% of data lies within 1 standard deviation of the mean
- Approximately 95% lies within 2 standard deviations
- Approximately 99.7% lies within 3 standard deviations
A normal distribution with mean 0 and standard deviation 1. Any normal distribution can be standardised using: z = (x − μ) ÷ σ
To find probabilities from a normal distribution, standardise to find the z-score, then use the standard normal tables provided in the exam. Remember: P(Z > z) = 1 − P(Z < z).