Summary

The whole chapter in one scroll — read this first, then dive into the detail below.

• A continuous random variable has infinitely many possible values, so we use a probability density function, f(x), instead of a probability mass function.
• Two rules make something a valid p.d.f: it's never negative, and the total area under it equals 1.
• Probabilities are found by integrating f(x) between two limits — the area under the curve.
• The cumulative distribution function, F(x), is the "running total" of probability up to a point: F(x) = P(X ≤ x).
• Mean and variance for continuous variables use the same logic as discrete ones, just with integrals instead of sums.
• The mode is where the graph peaks; the median is where F(x) = 0.5; comparing them tells you the skew.
• The continuous uniform distribution U[a, b] is the "flat rectangle" special case — every value in the range is equally likely.

1 · What Even Is a Continuous Random Variable?

Setting the foundation before any formulas show up.

Think about height. If I asked "what's the probability someone is exactly 175.0000000... cm tall?" — the honest answer is basically zero, because height can take an infinite number of decimal values. There's always a "more precise" measurement hiding underneath. This is the core difference from something like rolling a die, where the outcomes are countable and separate (1, 2, 3, 4, 5, 6).

Because any single exact value has probability zero, we stop asking "what's the chance of exactly this value?" and instead ask "what's the chance the value falls in this range?" That's what the probability density function, f(x), is built for — it doesn't give you a probability directly, it gives you a "density" that you turn into a probability by finding an area underneath it.

2 · The Cumulative Distribution Function

The "running total" version of the p.d.f.

Imagine you had to find P(1 < X < 3), then P(2 < X < 5), then P(0.5 < X < 4) — integrating from scratch every single time would get tedious fast. The cumulative distribution function, F(x), solves this by pre-calculating "the probability of being less than or equal to x" for every value of x, so you can just plug numbers in afterwards, like a lookup table.

3 · Mean and Variance

Same ideas as discrete random variables — swap sums for integrals.

If you've done discrete distributions, this will feel very familiar. Where a discrete mean is Σxp(x), the continuous mean simply replaces the sum with an integral and the probability mass function with the density function. The logic — "multiply each value by how likely it is, then add it all up" — hasn't changed at all.

4 · Mode, Median, Percentiles & Skewness

Three ways to describe "the middle" — and what happens when a distribution leans one way.

A The mode

The mode is simply the x-value where f(x) is at its largest — the "peak" of the curve, or the most densely-packed value. To find it, differentiate f(x), set the derivative to 0, and solve — exactly like finding a maximum point in regular calculus. (If f(x) is a straight increasing or decreasing line rather than a curve, the mode is just found by inspection at whichever endpoint gives the largest value — no differentiation needed.)

B The median and percentiles

The median is the value that splits the area under the curve exactly in half — 50% of the probability lies below it, 50% above. Because the c.d.f. already tells you "probability up to this point," finding the median is just a matter of solving F(m) = 0.5 for m. Quartiles and percentiles work exactly the same way, just with a different target value.

C Skewness

Skewness describes whether a distribution leans towards the small values or the large values.

• Positive skew ("bottom heavy") — the peak (mode) is at a lower x-value, and there's a long tail stretching out to the right, towards higher values. Ordering: mode < median < mean.
• No skew — the distribution is symmetrical (mode = median = mean).
• Negative skew ("top heavy") — the peak is at a higher x-value, with a long tail stretching out to the left. Ordering: mean < median < mode.

5 · The Continuous Uniform Distribution

The "flat rectangle" special case — every outcome equally likely.

This is the continuous equivalent of rolling a fair die — except instead of a few discrete outcomes, any value between two limits, a and b, is equally likely. Picture a perfectly flat, horizontal line between a and b — that flat line is the p.d.f. Since the area under it must equal 1, and the "rectangle" has width (b − a), the height must be 1/(b − a).

What to Memorise

Your final-hour, formula-sheet-level reference.

Concepts Checklist

Tick these off honestly — if you hesitate on any, go back and re-read that section.

• I can explain why continuous variables need a density function instead of exact probabilities.
• I can check whether a given function is a valid p.d.f. (positivity + total area = 1).
• I can find P(a < X < b) by integrating f(x).
• I can derive a c.d.f. from a p.d.f., including piecewise functions, and write it "in full."
• I can use a c.d.f. to find probabilities without re-integrating.
• I can find E(X) and Var(X) using the correct integral formulas (remembering the extra x or x²).
• I can apply E(aX+b) and Var(aX+b) without integrating again.
• I can find the mode by differentiating f(x) (or by inspection for linear functions).
• I can find the median/quartiles/percentiles by solving F(x) = target value.
• I can compare mean, median and mode to correctly identify positive/negative/no skew.
• I know the E(X), Var(X) and F(x) formulas for the continuous uniform distribution U[a,b] by heart.
• I can recognise real-world uniform distribution contexts (waiting times, rounding errors).

Exam Tips

Where marks are actually lost — and how to protect yours.