Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.7
📋 Learning Intentions
- Define a discrete random variable and state the conditions for a valid probability distribution.
- Construct a probability distribution table from a described random experiment.
- Calculate the expected value \( E(X) \), variance \( \text{Var}(X) \), and standard deviation \( \sigma \) of a discrete random variable.
- Apply the properties of expected value and variance to linear transformations of the form \( aX + b \).
- Interpret \( E(X) \) and \( \text{Var}(X) \) in context, distinguishing between a theoretical long-run average and an observed sample mean.
- Use a probability distribution to find probabilities of events defined by inequalities.
Real-World Introduction
An insurance company needs to set premiums for a travel policy. They know from historical data that 2% of policyholders will make a claim of $5 000, 5% will make a claim of $1 000, and the remaining 93% will make no claim. Before setting a price, the company must answer one question: on average, how much will each policy cost to pay out?
This is not a question about one specific customer — it is a question about the long-run average payout across thousands of customers. The mathematical tool that answers it is the expected value of a discrete random variable. The same tool is used in game design, quality control, genetics, and financial modelling wherever outcomes are uncertain but their probabilities are known.
🎲 Analogy — The Weighted Average Think of \( E(X) \) as a weighted average: each outcome is "weighted" by how likely it is. A fair die has \( E(X) = 3.5 \) — not because 3.5 ever appears on the die, but because, over many rolls, the average result converges to 3.5.
Core Concepts & Definitions
When an experiment has numerical outcomes that depend on chance, it is useful to assign a variable to represent those outcomes. Unlike an ordinary variable in algebra, this variable does not have one fixed value — its value is determined by the result of a random process.
📖 Key Definition — Discrete Random Variable A discrete random variable \( X \) is a variable whose value is determined by the outcome of a random experiment, and which can take only a countable set of distinct numerical values. Each value \( x_i \) has an associated probability \( P(X = x_i) \).
The word discrete means the values are separated