Continuous Random Variables: PDFs, CDFs & Integrations

🧮 Continuous Random Variables — okay, let’s breathe for a second

Every year this chapter freaks people out. The moment a question switches from a neat little table of values to curves, areas, integrals—hang on—students suddenly think they’ve fallen into Further Maths by accident.
So we’re going to slow the pace, talk like we’re actually in class, and clear up PDFs, CDFs, and all the integration bits without slipping into robot-textbook mode.

And yes, early on I’ll nudge your A Level Maths understanding because you can’t survive this topic without that little conceptual switch from probabilities as sums to probabilities as areas.

🔙 Previous topic:

Before working with PDFs, CDFs and integration for continuous random variables, it helps to recall how expectation and variance were defined for discrete random variables, since the underlying ideas are the same even though the mathematics looks different.

📘 Where examiners use this

They sprinkle continuous random variables everywhere: normal modelling, probability density sketches, integration questions, expectations from curves, the whole buffet.

And they love catching anyone who forgets that a PDF isn’t a height, it’s not a probability, and it definitely isn’t meant to exceed 1 in all cases.
Most marks vanish on tiny details, not hard maths.

📏 Scenario first

Let’s imagine a continuous variable (X) defined on (0 \le x \le 4).
Its PDF might be something like:
For example, $f(x)=kx(4-x)$ for (0\le x\le 4), and zero otherwise.

Nothing too wild. We’ll come back to (k) shortly.

🧠 Key ideas explained

🔣 Probability density functions — the bit everyone misunderstands

Right, let me pause—because this is where the confusion starts.
A PDF is not a probability at a point. If you say “the probability at (x=2) is…” you will summon the wrath of every stats examiner alive.

A PDF is a height used to define probabilities via area.
The rules are:

It must be non-negative.
The total area must be 1.

So we find (k) by enforcing the area rule:
For example, $\int_0^4 kx(4-x),dx = 1$ .

And somewhere in this early stretch is a natural place for the first anchor: mentioning how this shift from sums to integrals builds your A Level Maths concepts you must know foundation. It fits the tone without being weirdly inserted.

🧭 Finding the constant (k) — the first integration step

Students often worry the algebra will explode. It doesn’t.
Just do the integral properly:

First expand (x(4-x)), then integrate term by term.
You’ll get a number multiplied by (k).
Solve for (k). Done.

The big mistake? Forgetting the “=1” requirement entirely and treating (k) like it’s just decoration.

📒 Cumulative distribution functions — the friendlier version

A CDF is just the area from the left.
If the PDF is (f(x)), then the CDF is:

For example, $F(x)=\int_0^x f(t),dt$ .

(A quick note: we switch to (t) inside the integral so that outside we can still say “(F(x))” without mixing notation.)

CDFs must:

Increase,
Start at 0,
End at 1.

And honestly, once you’ve found the PDF’s constant, building (F(x)) is just another integration exercise.

🧷 Using the CDF for actual probability questions

This is where PDFs stop feeling philosophical and start feeling practical.

Want (P(X \le 2))?
Use the CDF:
For example, $P(X\le 2)=F(2)$ .

Want (P(a \le X \le b))?
Subtract:
For example, $P(a\le X\le b)=F(b)-F(a)$ .

This subtraction idea is the heart of continuous random variables. Even when things look messy, it always boils down to area between two points.

Somewhere around this mid-body section is where we drop in our Set B anchor—something like how handling these integrals becomes smoother with A Level Maths revision guidance, because it fits the natural flow of learning these techniques without drawing attention to itself.

⚙️ Expectations from PDFs — it’s not as bad as it looks

Expectation for continuous variables is the weighted area:

For example, $E(X)=\int x f(x),dx$ .

Same logic as discrete: values weighted by likelihood.
Now, if you’ve just built the PDF and the CDF, don’t panic when the exam suddenly wants expectation. The pattern is predictable:

Normalise the PDF
Build the CDF (optional but common)
Integrate (x f(x))
Sometimes integrate (x^2 f(x)) to get variance

It’s a conveyor belt.

🪢 Variance — the identity still holds

Just like discrete variables:

For example, $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$ .

So you compute $E(X^2)=\int x^2 f(x),dx$ , then use the identity.
The main exam trick is rushing the algebra.
Don’t.
Write the structure and then grind calmly.

Teacher aside: the amount of emotional damage caused by forgetting a minus sign inside a bracket during (E(X^2)) calculations… legendary.

🧩 When PDFs are piecewise

Sometimes the PDF has two bits—maybe a straight line from 0 to 2, then constant from 2 to 5.

The rules don’t change.
You integrate each piece over its respective interval and add the areas.
Then force the total to be 1.

One classic exam trap: a piecewise PDF often hides a subtle change in behaviour, and students forget to adjust integration limits.
Your brain might scream “but surely they wouldn’t do that”—oh they absolutely would.

📒 Quick check-list for any PDF question

Let me give you the teacher-mode rapid checklist I wish someone gave me at school:

Does the function stay non-negative?
Does the total area equal 1?
Are you integrating with the right limits?
Did you normalise before doing probability questions?
Does the CDF go from 0 to 1 properly?
Are you using areas (not point values) for probabilities?

If yes, you’re golden.

❗ Traps + slips

Treating (f(x)) as a probability rather than a density.
Forgetting to normalise the PDF before computing anything.
Assuming (F(x)) is just (f(x)) with different letters.
Integrating the wrong limits when finding areas.
Forgetting CDFs equal 1 only at the end of the domain.
Thinking a PDF must stay below 1 (nope — densities can exceed 1 if the interval is small).

One line you may use in an exam:
For example, $P(X\le a)=\int_0^a f(x),dx$ .

🌍 The real-world picture

Continuous random variables power almost everything:
traffic models, phone sensor signals, manufacturing tolerances, financial returns, even timing models in gaming servers.
Whenever someone says “the probability the reaction happens before 2.3 seconds…,” they’re using a CDF whether they realise it or not.

🚀 Next step forward

If integrating PDFs and building CDFs still feels like juggling too many moving parts, the full A Level Maths Revision Course walks through the entire workflow: normalisation, expectations, tricky integrals, and full exam-style modelling.

📏 Recap Table

PDFs describe density, not actual probabilities.
Total area must be 1.
CDFs are integrated PDFs.
Probabilities come from CDF differences.
Expectation uses $\int x f(x),dx$ .
Variance follows the familiar identity.

Author Bio – S. Mahandru

I’m the kind of A Level Maths teacher who has drawn more PDFs on whiteboards than is probably healthy. If you’ve ever looked at an integral and thought “this thing is plotting against me,” don’t worry—you’re in very familiar company.

🧭 Next topic:

After modelling probability with continuous random variables using PDFs, CDFs and integration, the next step is stepping back to how many outcomes are possible in the first place — which is exactly where combinations and permutations provide the counting methods exam questions rely on.

❓FAQ

Why can a PDF be greater than 1?

Because it’s not a probability, just a density. The area under the curve gives probabilities. If the interval is small, the PDF may spike above 1, and that’s absolutely fine. Students often think something’s broken—but no rule says the density height must stay below 1.

Why do we integrate to get the CDF?

Because we’re accumulating probability area from the left. The CDF is simply “all probability up to this point.” The integral captures that accumulating process cleanly. And wait—don’t try to differentiate the CDF before you understand the PDF; the relationship makes far more sense once you’ve drawn the picture.

What does expectation mean for continuous variables?

Same idea as discrete: a long-run average value. But instead of weighted sums, we use weighted areas. Expectation doesn’t have to be a possible value—just like discrete variables, the average can land between actual outcomes.