Discrete random variables, the binomial and normal distributions

Learning objectives

Construct the probability distribution of a discrete random variable, check that the probabilities sum to 1, calculate the expected value E(X) = Σ x P(X = x), interpret it as the long-run average outcome and decide whether a game is fair by testing whether the expected gain is zero

Recognise a binomial situation by its fixed number of independent trials with a constant probability of success, write X ~ B(n, p), and calculate P(X = r), P(X ≤ r) and P(X ≥ r) with technology

Calculate the mean np and the variance np(1 − p) of a binomial distribution and interpret them in context

Describe the normal curve as symmetric about the mean with mean, median and mode equal, and sketch it with the required region shaded before any calculation

Calculate normal probabilities with technology for X ~ N(μ, σ²), and use the inverse normal function to find the value of x corresponding to a given cumulative probability

Use the facts that about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean to check normal calculations, knowing that μ and σ are always given for an inverse normal calculation (finding an unknown μ or σ with z-values is not part of this course)