6.3 Continuous Random Variables

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand what a continuous random variable is, and recall and use the properties of a probability density function (PDF) — including for functions defined over infinite domains such as f(x) = 3/x⁴ for x ≥ 1.
  2. Use a PDF to find probabilities, and to calculate the mean and variance of a distribution — including finding the median and other percentiles by working directly with areas under the curve. (Note: you do not need to know about the cumulative distribution function.)

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Part 1 — Discrete vs Continuous Random Variables

You have already studied discrete random variables — these are variables that can only take specific, separate values, like the number you roll on a die (1, 2, 3, 4, 5, or 6). You can list every possible value.

A continuous random variable is different. It can take any value within a range, not just whole numbers or set values. Examples include:

  • The height of a person (could be 170.1 cm, 170.15 cm, 170.153 cm, etc.)
  • The time it takes to complete a task
  • The lifetime of a light bulb in years

Because there are infinitely many possible values, we cannot list them all or assign a probability to each individual value. Instead, we use a probability density function (PDF) to describe the distribution.

Key idea: For a continuous random variable, the probability that X equals any single exact value is always zero. We can only find the probability that X lies within an interval.

Part 2 — Probability Density Functions (PDFs)

What is a PDF?

A probability density function, written f(x), is a mathematical function (a formula or equation) that tells you how the probability is spread out across different values of a continuous random variable X.

Think of it like a shape. The area under the curve of f(x) between two values represents the probability that X lies between those values.

The Two Essential Properties of a Valid PDF

For f(x) to be a valid probability density function, it must satisfy both of the following:

Property 1 — Non-negativity:

f(x)≥0for all values of xf(x) \geq 0 \quad \text{for all values of } x

The function must never be negative. A negative probability makes no sense.

Property 2 — Total area equals 1: ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1

In practice, if f(x) is only defined over an interval [a, b] and equals zero everywhere else, this becomes: ∫abf(x) dx=1\int_a^b f(x) \, dx = 1

The total probability must add up to 1 (certainty). Because the PDF covers all possibilities, the total area under the curve must always equal exactly 1.

PDFs Over Infinite Domains

The syllabus includes PDFs defined over infinite ranges. For example:

f(x)=3x4for x≥1,f(x)=0 otherwisef(x) = \frac{3}{x^4} \quad \text{for } x \geq 1, \quad f(x) = 0 \text{ otherwise}

Even though the domain goes to infinity, the area can still equal 1. To check:

∫1∞3x4 dx=[−1x3]1∞=0−(−1)=1✓\int_1^{\infty} \frac{3}{x^4} \, dx = \left[-\frac{1}{x^3}\right]_1^{\infty} = 0 - (-1) = 1 \checkmark

This is a valid PDF because: (1) f(x) ≥ 0 for all x ≥ 1, and (2) the total area equals 1.

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