Why is the probability of a continuous random variable taking a single value, P(X=c), equal to zero?

Because probability is represented by the area under the PDF curve. The area of a region with zero width (at a single point 'c') is zero. Think of it as integrating from c to c: $\int_{c}^{c} f(x) dx = 0$.

What is the difference between a PDF, f(x), and a CDF, F(x)?

The PDF, $f(x)$, is the 'density' function; its value is not a probability, but the area under it is. The CDF, $F(x)$, is a cumulative probability function; its value at $x$ is the actual probability $P(X \le x)$. The PDF is the derivative of the CDF.

My variance calculation gave a negative number. What did I do wrong?

Variance can never be negative. If you get a negative answer, you have made a calculation error. The most common mistakes are: (1) an error in the integration of $x^2 f(x)$, or (2) forgetting to square $E(X)$ in the formula $Var(X) = E(X^2) - [E(X)]^2$. The worked example above shows how to spot and correct this.

How do I define the CDF for a piecewise PDF?

You need to define $F(x)$ for all real numbers. It will be 0 for $x$ less than the lower bound of the PDF's range. It will be 1 for $x$ greater than the upper bound. In between, you integrate the PDF from the lower bound up to a variable $x$. If the PDF has multiple pieces, the CDF will also have multiple corresponding pieces.

Where can I learn Continuous random variables for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 6.3 (Continuous random variables) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/6-3-continuous-random-variables.

Is Continuous random variables (6.3) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Continuous random variables.

Continuous random variables 9709 — Free…

Continuous random variables

In Probability & Statistics 1, we dealt with discrete random variables, which can only take specific, separate values (like the score on a dice). Now, we explore continuous random variables, which can take any value in a given range, such as the height of a student or the time taken to run a race. Instead of a probability mass function, we use a probability density function (PDF), where probabilities are represented by areas.

BY THE END, YOU CAN…

1Understand the properties of a probability density function (PDF) and use it to calculate probabilities.
2Understand the cumulative distribution function (CDF) and its relationship with the PDF.
3Calculate the expectation (mean) and variance of a continuous random variable by integration.
4Solve problems involving continuous random variables, including finding unknown constants in a PDF.

In simple terms

A friendly intro before the formal notes — no formulas yet.

From Density to Probability

Unlike discrete variables which have specific probabilities, continuous variables use a 'probability density function' (PDF) where probability is the area under the curve. We use integration to find these areas and other key measures like the mean and variance.

THINK OF IT LIKE…

Imagine pouring one litre of sand onto a narrow strip of ground. The height of the sand at any point is the 'density' (our PDF). The total amount of sand is fixed at one litre (total probability is 1). To find the amount of sand in a specific section, you'd measure the volume in that section (calculating the area under the curve), not just the height at one single point.

1
The PDF, f(x), shows the relative likelihood of a value. It must be non-negative, and the total area under its curve over its entire range must equal 1.
2
The CDF, F(x), gives the total probability up to a value x, P(X ≤ x). It's the cumulative area under the PDF. The probability between two points, P(a < X ≤ b), is simply the difference F(b) − F(a).
3
The mean E(X) and variance Var(X) are found by integrating over the variable's range. E(X) is the integral of x⋅f(x), and Var(X) is found using the integral of x²⋅f(x).
4
A common example is the uniform distribution, where the PDF is a constant height over a fixed interval. Its graph is a rectangle where height times width equals 1.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$

Tap a symbol — great for exam definitions

$F(x) = P(X \le x) = \int_{a}^{x} f(t) dt$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

The Probability Density Function (PDF)

The probability density function, denoted $f(x)$ , describes the relative likelihood for a continuous random variable to take on a given value. Unlike for discrete variables, $f(x)$ is not a probability itself. Instead, the area under the graph of $f(x)$ over an interval corresponds to the probability that the variable falls within that interval.

For $f(x)$ to be a valid PDF, it must satisfy two conditions.
Condition 1: $f(x) \ge 0$ for all values of $x$ . The graph can never be below the x-axis.
Condition 2: The total area under the curve must be 1. $\int_{-\infty}^{\infty} f(x) dx = 1$ . In practice, we integrate over the interval where $f(x)$ is non-zero.

Expectation and Variance

Just as with discrete variables, we can calculate the mean (or expectation) and variance for continuous random variables. The expectation, $E(X)$ , is the long-run average value of X, and the variance, $Var(X)$ , measures the spread of the distribution. Instead of using summation ($Sigma$), we now use integration.

Expectation: $E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$

Variance: $Var(X) = \sigma^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - [E(X)]^2$

EXAM TIP

Always use the full defined range of $x$ for the limits of integration when calculating $E(X)$ and $E(X^2)$ . A common error is to use the limits from a probability calculation (e.g., $P(X>1)$ ) instead of the full range (e.g., 0 to 2).

The Cumulative Distribution Function (CDF)

The cumulative distribution function, $F(x)$ , gives the probability that the random variable $X$ is less than or equal to a particular value $x$ . It is the integral of the PDF from the lower bound of the range up to $x$ . A key property is that the derivative of the CDF is the PDF: $F'(x) = f(x)$ .

$F(x) = P(X \le x) = \int_{a}^{x} f(t) dt$ (where $a$ is the start of the range)

$P(c < X < d) = F(d) - F(c)$

The CDF is a non-decreasing function, starting at 0 and ending at 1.
For a piecewise PDF, the CDF will also be piecewise. You must define it for all real values of $x$ .
For $x$ below the range of the PDF, $F(x)=0$ . For $x$ above the range, $F(x)=1$ .

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A continuous random variable X has a probability density function given by $f(x) = k(4-x^2)$ for $0 \le x \le 2$ , and $f(x) = 0$ otherwise.

(i) Show that $k = \frac{3}{16}$ . (ii) Find $P(X > 1)$ .

1
(i) For $f(x)$ to be a valid PDF, the total area under the curve must be 1. We integrate $f(x)$ over its defined range and set the result to 1.

EXAMPLE 2

For the random variable X with PDF $f(x) = \frac{3}{16}(4-x^2)$ for $0 \le x \le 2$ , find $E(X)$ and $Var(X)$ .

1
First, find the expectation, $E(X)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAContinuous random variables

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a continuous random variable?

TAP OR SPACE TO FLIP

ANSWER

A random variable that can take any value within a given range. Examples include height, weight, or time.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
For $f(x)$ to be a valid PDF, it must satisfy two conditions.
✓
Condition 1: $f(x) \ge 0$ for all values of $x$ . The graph can never be below the x-axis.
✓
Condition 2: The total area under the curve must be 1. $\int_{-\infty}^{\infty} f(x) dx = 1$ . In practice, we integrate over the interval where $f(x)$ is non-zero.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Continuous Random Variable Problems

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Continuous Random Variable Problems on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS6.2 Linear combinations of random variables NEXT →6.4 Sampling and estimation

Continuous random variables

BY THE END, YOU CAN…

1Understand the properties of a probability density function (PDF) and use it to calculate probabilities.
2Understand the cumulative distribution function (CDF) and its relationship with the PDF.
3Calculate the expectation (mean) and variance of a continuous random variable by integration.
4Solve problems involving continuous random variables, including finding unknown constants in a PDF.

In simple terms

A friendly intro before the formal notes — no formulas yet.

From Density to Probability

THINK OF IT LIKE…

1
The PDF, f(x), shows the relative likelihood of a value. It must be non-negative, and the total area under its curve over its entire range must equal 1.
2
The CDF, F(x), gives the total probability up to a value x, P(X ≤ x). It's the cumulative area under the PDF. The probability between two points, P(a < X ≤ b), is simply the difference F(b) − F(a).
3
The mean E(X) and variance Var(X) are found by integrating over the variable's range. E(X) is the integral of x⋅f(x), and Var(X) is found using the integral of x²⋅f(x).
4
A common example is the uniform distribution, where the PDF is a constant height over a fixed interval. Its graph is a rectangle where height times width equals 1.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$

Tap a symbol — great for exam definitions

$F(x) = P(X \le x) = \int_{a}^{x} f(t) dt$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

The Probability Density Function (PDF)

For $f(x)$ to be a valid PDF, it must satisfy two conditions.
Condition 1: $f(x) \ge 0$ for all values of $x$ . The graph can never be below the x-axis.
Condition 2: The total area under the curve must be 1. $\int_{-\infty}^{\infty} f(x) dx = 1$ . In practice, we integrate over the interval where $f(x)$ is non-zero.

Expectation and Variance

Expectation: $E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$

Variance: $Var(X) = \sigma^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - [E(X)]^2$

EXAM TIP

The Cumulative Distribution Function (CDF)

$F(x) = P(X \le x) = \int_{a}^{x} f(t) dt$ (where $a$ is the start of the range)

$P(c < X < d) = F(d) - F(c)$

The CDF is a non-decreasing function, starting at 0 and ending at 1.
For a piecewise PDF, the CDF will also be piecewise. You must define it for all real values of $x$ .
For $x$ below the range of the PDF, $F(x)=0$ . For $x$ above the range, $F(x)=1$ .

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A continuous random variable X has a probability density function given by $f(x) = k(4-x^2)$ for $0 \le x \le 2$ , and $f(x) = 0$ otherwise.

(i) Show that $k = \frac{3}{16}$ . (ii) Find $P(X > 1)$ .

1
(i) For $f(x)$ to be a valid PDF, the total area under the curve must be 1. We integrate $f(x)$ over its defined range and set the result to 1.

EXAMPLE 2

For the random variable X with PDF $f(x) = \frac{3}{16}(4-x^2)$ for $0 \le x \le 2$ , find $E(X)$ and $Var(X)$ .

1
First, find the expectation, $E(X)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAContinuous random variables

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a continuous random variable?

TAP OR SPACE TO FLIP

ANSWER

A random variable that can take any value within a given range. Examples include height, weight, or time.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
For $f(x)$ to be a valid PDF, it must satisfy two conditions.
✓
Condition 1: $f(x) \ge 0$ for all values of $x$ . The graph can never be below the x-axis.
✓
Condition 2: The total area under the curve must be 1. $\int_{-\infty}^{\infty} f(x) dx = 1$ . In practice, we integrate over the interval where $f(x)$ is non-zero.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Continuous Random Variable Problems

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Continuous Random Variable Problems on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS6.2 Linear combinations of random variables NEXT →6.4 Sampling and estimation

Revise Continuous random variables for Cambridge 9709 (6.3)

In simple terms

Explore the concept

Key formulas

Full topic notes

The Probability Density Function (PDF)

Expectation and Variance

The Cumulative Distribution Function (CDF)

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Continuous Random Variable Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

The Probability Density Function (PDF)

Expectation and Variance

The Cumulative Distribution Function (CDF)

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Continuous Random Variable Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Continuous random variables