Why P(X = a) is exactly zero, why a density can exceed 1, and the variance shortcut.
Ask what P(X = 1) is before you shrink anything.
Most will say "small" or will read off f(1) = 0.375. Shrinking the strip drives the probability to zero while the red marker at 0.375 never moves, so the two quantities visibly come apart.
Watch the third readout as it goes: probability divided by width settles on 0.375. That is what the word density means, and it arrives here as an observation rather than a definition.
f(x) = 2 on [0, 0.5] is a perfectly legitimate density, because 2 × 0.5 = 1. No probability is ever greater than 1, so a density that exceeds 1 settles the question permanently.
And P(X < a) = P(X ≤ a) for continuous variables, because the single point contributes nothing. For discrete variables they differ, and questions use both deliberately.
| 1. Var(X) from the table | 6.6 − 2.4² = 6.6 − 5.76 = 0.84. |
| 2. P(X < 1) | [x³/8] from 0 to 1 = 0.125. |
| 3. P(X = 1.5) | B, exactly 0. A point encloses no area. |
1 markUsing ∫ f = 1 to find an unknown constant, which is usually part (a) and everything else depends on it.
1 markLimits taken from the question, not from the whole domain.
1 markE(X²) − [E(X)]², with both quantities shown separately.
| They give | What it means |
|---|---|
| 4.2 (Q1) | Subtracted E(X) rather than its square. |
| −0.84 (Q1) | Reversed the order. A negative variance is impossible, which is the self-check to teach. |
| 0.917 (Q1) | Gave the standard deviation. Fine mathematics, wrong question. |
| 0.375 (Q2) | Gave the density at x = 1 rather than the area to its left. This is the exact confusion the figure targets. |
| 0.5 (Q2) | Assumed half the range is half the probability. The density is far larger near 2, which the picture shows. |
| "A small positive number" (Q3) | Nearly there, and worth praising before correcting: they know it is tiny, they have not yet accepted it is exactly zero. |
"If every value has probability zero, how does X take one?" A genuinely good question and worth sitting with. Zero probability does not mean impossible for a continuous variable; probability lives in intervals, not points.
"Can f be bigger than 1?" Yes, as long as the total area is 1. A narrow, tall density is exactly how a precise measurement looks.
"Do I integrate or sum?" Continuous integrates, discrete sums. The structure is identical otherwise, which is worth drawing side by side.
| What is happening | |
|---|---|
| 1 | Predict P(X = 1). Shrink the strip. Let the two readouts separate. |
| 2 | The conditions on a density, and the f = 2 example that exceeds 1. |
| 3 | E(X) and Var(X) by integration, on this density, by hand. |
| 4 | The discrete table, with the shortcut and the long way compared. |
| 5 | A find-the-constant question, which is how this is nearly always examined. |
Do not say P(X = a) is "very small". It is zero, and "very small" leaves the misconception fully intact.
Do not introduce the density and the distribution function in the same lesson. One idea at a time; the cdf is a better second lesson than a worse first one.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.