Continuous Random Variables and PDFs

A continuous random variable is a quantity whose value can take any real number within some interval, rather than only a finite or countable set of values. Unlike a discrete random variable (such as the number of heads in ten coin flips), a continuous random variable might represent the exact height of a randomly selected student, the time until a radioactive atom decays, or the temperature at noon tomorrow. Because there are infinitely many possible values, the probability of landing on any single exact value is zero — what matters instead is the probability of falling within a range. To describe how probability is spread across those infinitely many values, mathematicians use a probability density function (PDF), often written f(x). The PDF is a non-negative function whose total area under the curve equals exactly 1. The probability that the random variable X falls between two values a and b is the area under f(x) from a to b, computed as a definite integral. This shifts the language of probability from simple counting to integral calculus. Once you have a PDF, you can compute essential summary statistics: the mean (expected value) μ tells you the 'centre of mass' of the distribution, the variance σ² measures how spread out the values are around that centre, and the standard deviation σ is the square root of the variance and shares the same units as X. All three are computed using integration, making calculus the natural language of continuous probability.

Key takeaways
  • What makes a valid PDF?
  • Computing probabilities as areas
  • The cumulative distribution function (CDF)
  • Expected value (mean)
  • Variance and standard deviation

What makes a valid PDF?

A function f(x) is a valid probability density function if it satisfies two conditions: (1) f(x) ≥ 0 for all x, and (2) the integral over all possible values equals 1, i.e. ∫_{-∞}^{∞} f(x) dx = 1. The first condition ensures probabilities are never negative. The second ensures total probability is 100%. If a candidate function fails either condition, it cannot serve as a PDF.

Computing probabilities as areas

For a continuous random variable X with PDF f(x), the probability that X falls in the interval [a, b] is P(a ≤ X ≤ b) = ∫_a^b f(x) dx. Because a single point has zero width, P(X = c) = 0 for any specific value c. This means P(a ≤ X ≤ b) = P(a < X < b) — whether endpoints are included makes no difference for continuous distributions.

The cumulative distribution function (CDF)

The cumulative distribution function F(x) = P(X ≤ x) = ∫_{-∞}^{x} f(t) dt gives the total probability accumulated up to the value x. By the Fundamental Theorem of Calculus, f(x) = F'(x) whenever f is continuous. The CDF is always non-decreasing, starts at F(-∞) = 0, and ends at F(+∞) = 1. You can find P(a ≤ X ≤ b) quickly as F(b) − F(a).

Expected value (mean)

The mean or expected value of X is μ = E[X] = ∫_{-∞}^{∞} x · f(x) dx. Think of it as a continuous weighted average: each possible value x is weighted by how much probability density f(x) sits at that value. If the distribution is symmetric about some centre point, the mean equals that centre. For a non-symmetric (skewed) distribution, the mean is pulled toward the heavier tail.

Variance and standard deviation

The variance is σ² = Var(X) = E[(X − μ)²] = ∫_{-∞}^{∞} (x − μ)² · f(x) dx. A computationally useful equivalent formula is σ² = E[X²] − μ², where E[X²] = ∫_{-∞}^{∞} x² · f(x) dx. Variance is in squared units of X, so the standard deviation σ = √(Var(X)) is taken to restore the original units and give an interpretable measure of spread.

The Key Equation

μ=∫−∞∞x f(x) dx,σ2=∫−∞∞(x−μ)2 f(x) dx\mu = \int_{-\infty}^{\infty} x\, f(x)\, dx, \qquad \sigma^2 = \int_{-\infty}^{\infty} (x-\mu)^2\, f(x)\, dx
  • \mu — Mean (expected value) (same as X): The probability-weighted average of all possible values of X; the 'centre of mass' of the distribution.
  • \sigma^2 — Variance (square of units of X): The probability-weighted average of squared deviations from the mean; measures the spread of the distribution.
  • f(x) — Probability density function (per unit of X): A non-negative function describing the relative likelihood of X taking a given value; area under curve equals 1.
  • \sigma — Standard deviation (same as X): The square root of the variance, returning spread to the original units of measurement.

Worked Example

Problem. A continuous random variable X has PDF f(x) = 3x² for 0 ≤ x ≤ 1 (and 0 elsewhere). (a) Verify that f is a valid PDF. (b) Find P(0.5 ≤ X ≤ 1). (c) Find the mean μ. (d) Find the variance σ² and standard deviation σ.

P(0.5≤X≤1)=0.875,μ=34,σ2=380,σ=380≈0.194P(0.5 \le X \le 1)=0.875,\quad \mu=\tfrac{3}{4},\quad \sigma^2=\tfrac{3}{80},\quad \sigma=\sqrt{\tfrac{3}{80}}\approx 0.194

Answer. f is valid; P(0.5 ≤ X ≤ 1) = 0.875; μ = 0.75; σ² = 3/80 ≈ 0.0375; σ ≈ 0.194

Figure
Diagram
PDF: f(x) = 3x² on [0,1]xf(x)00.5103x=0.5μ=0.75shaded area= 0.875f(x)=3x²
Origins · 1 minAndrey Kolmogorov and how this was discovered
AK
Andrey Kolmogorov
1903–1987 · Russian

Kolmogorov placed probability theory on a rigorous mathematical foundation with his 1933 monograph 'Grundbegriffe der Wahrscheinlichkeitsrechnung' (Foundations of Probability). He introduced the axiomatic framework — including the concept of a probability measure and the sigma-algebra — that made it possible to define continuous random variables and their PDFs with complete mathematical precision. Without his axioms, continuous probability would lack the logical foundation needed to apply integration so confidently. Kolmogorov was also a leading contributor to turbulence theory, algorithmic complexity, and mathematical logic — making him one of the most broadly influential mathematicians of the twentieth century.

  • Discrete Random Variables and PMFs — Discrete random variables use probability mass functions (PMFs) and sums instead of integrals; understanding the contrast clarifies why PDFs and integration are needed for the continuous case.
  • The Normal Distribution — The normal distribution is the most important continuous distribution, defined entirely by its PDF f(x) = (1/σ√(2π)) e^{−(x−μ)²/(2σ²)}, and uses the mean and variance concepts developed here.
  • Definite Integrals and Area Under a Curve — All probability computations for continuous random variables reduce to evaluating definite integrals; fluency with integration techniques is essential for this topic.
  • Cumulative Distribution Functions (CDF) — The CDF F(x) is the antiderivative of the PDF and provides an efficient way to compute probabilities over intervals without re-integrating f(x) each time.