| Term | Meaning / quick note |
|---|---|
| Discrete random variable (X) | A variable that takes a finite or countable set of values (for example, 1,2,3,4,5). We attach probabilities to each possible value. |
| Probability mass function (pmf) | The rule P(X = x) giving the probability for each x in the support. Probabilities must be ≥ 0 and sum to 1. |
| Support | The set of x values for which P(X = x) > 0 (e.g., {1,2,3,4,5}). |
| Expected value E(X) | Average or long-run mean: E(X) = Σ x·P(X = x). Think of it as the weighted average of outcomes using probabilities as weights. |
| Variance Var(X) | Measure of spread: Var(X) = E(X2) − [E(X)]2. Standard deviation is √Var(X). |
| Fair game | If E(X) = 0 (player’s expected gain is zero) the game is fair; positive E(X) favours the player, negative E(X) favours the house. |
📌 1. Probability distribution for a discrete random variable
What you are given
- Pointer 1 — The pmf: The distribution is given as a table or formula P(X = x) = …; check that probabilities are non-negative and sum to 1 (Σ P(X = x) = 1).
- Pointer 2 — Support: Identify the set of possible x values (for example {1,2,3,4,5}). Always keep the support visible when you compute sums — it avoids mistakes.
- Pointer 3 — Units & context: State what X measures (units, currency, points). This is essential when you interpret E(X) later.
🌍 Real-World Connection
Discrete distributions appear when outcomes are naturally countable — for example, the number of heads in a few coin tosses, number of defective items in a batch, or prizes won in a single game. Always link probabilities to realistic mechanisms (dice, draws, surveys).
📌 2. Expected value: definition and intuition
Definition (calculation)
- Step: Compute E(X) = Σ x · P(X = x). Do the multiplication term-by-term and keep a running total — write intermediate results to avoid arithmetic slips.
- Interpretation: E(X) is the long-run average if the random experiment is repeated many times; it need not be a value that X can actually take.
Worked example — compute E(X), Var(X), interpretation
Given distribution:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | 0.10 | 0.20 | 0.15 | 0.05 | 0.50 |
Compute E(X):
- 1 × 0.10 = 0.10
- 2 × 0.20 = 0.40
- 3 × 0.15 = 0.45
- 4 × 0.05 = 0.20
- 5 × 0.50 = 2.50
Sum: E(X) = 0.10 + 0.40 + 0.45 + 0.20 + 2.50 = 3.65.
Compute Var(X):
- Compute E(X2) = Σ x2P(X = x):
- 12×0.10 = 0.10
- 22×0.20 = 0.80
- 32×0.15 = 1.35
- 42×0.05 = 0.80
- 52×0.50 = 12.50
E(X2) = 0.10 + 0.80 + 1.35 + 0.80 + 12.50 = 15.55.
Var(X) = E(X2) − [E(X)]2 = 15.55 − (3.65)2 = 15.55 − 13.3225 = 2.2275.
Standard deviation ≈ √2.2275 ≈ 1.49.
Interpretation (intuitive): The average outcome if the game were repeated many times is 3.65 units (for example currency), and individual outcomes typically deviate about 1.5 units from that average; because E(X) is positive here, the player expects a net gain on average.
🧠 Examiner Tip
- Write intermediate products (x·P) in your working: examiners award method marks for correctly listing these even if the final sum has a small arithmetic slip.
- When asked whether a game is fair, quote E(X) and explain in context: “E(X) = 3.65 (units) → not fair: player wins on average 3.65 per play”.
- Label units (currency, points) when you interpret E(X) and Var(X).
📌 3. Linearity & short tricks
- Linearity of expectation: E(aX + b) = a·E(X) + b. Use this to compute expected values of transformed variables without re-summing the whole pmf.
- Variance with constants: Var(aX + b) = a2Var(X). Adding a constant b does not change variance; scaling by a multiplies variance by a2.
- Check sums early: If Σ P = 1 fails, stop: the pmf is invalid — request or correct the distribution before computing E(X).
📱 GDC Use
- Enter values in two lists (X list and P list) and compute sumproduct(X,P) for E(X) and sumproduct(X2,P) for E(X2).
- Use the calculator to check ΣP = 1 quickly and to compute Var(X) accurately to several decimal places.
📌 4. Fair games, decision rules & interpretation
- Fairness rule: If E(X) = 0 the game is fair (player’s expected gain is zero). If E(X) > 0 the player expects to gain in the long run; if E(X) < 0 the house expects to profit on average.
- Risk vs expectation: E(X) only tells the average — variance/SD indicate risk. A high positive E(X) with huge variance may still be risky for a short sequence of plays.
- Practical decision: To decide if you should play once, weigh E(X) (expected gain) against Var(X) (risk) and your risk tolerance.
📐 IA Spotlight
Design an experiment to simulate the game many times (e.g., 10,000 runs) to estimate empirical E(X) and SD; compare the simulated means to theoretical E(X) and comment on convergence and sampling variability.
🔍 TOK Perspective
- Discuss whether the mathematical model (pmf) is an adequate representation of a real-world game — what assumptions are being made about randomness and independent trials?
- Is a game still ‘fair’ if E(X)=0 but the distribution has very large tails that impose practical risk to players?
📝 Paper Tips
- Always state the support set explicitly (e.g., X ∈ {1,2,3,4,5}) before summing; this helps examiners follow your method.
- Show the term-by-term products x·P(X=x) and the sum; method marks may be awarded for correct intermediate terms even if final addition has a small error.
- When asked whether the game is fair, write E(X) with units and a one-line conclusion: “E(X) = 3.65 units → the player expects to win on average 3.65 units per play; therefore the game is not fair.”
- If asked for variance, compute E(X2) first then apply Var(X) = E(X2) − [E(X)]2; record arithmetic carefully and box your final answers.
📌 Quick summary
- Check pmf validity (nonnegative probabilities, sum to 1).
- Compute E(X) as Σ x·P and interpret in context (long-run average).
- Compute Var(X) via E(X2) − [E(X)]2 and use SD to comment on spread/risk.
- Relate E(X) to fairness and decision making; always include units and a reasoned conclusion.