4.1 Continuous Random Variables

2026 Syllabus Objectives

  1. Use a probability density function which may be defined piecewise
  2. Use the general result E(g(X))=∫f(x)g(x)dxE(g(X)) = \int f(x)g(x)dx where f(x)f(x) is the probability density function of the continuous random variable XX and g(X)g(X) is a function of XX
  3. Understand and use the relationship between the probability density function (PDF) and the cumulative distribution function (CDF), and use either to evaluate probabilities or percentiles
  4. Use cumulative distribution functions (CDFs) of related variables in simple cases (e.g., given the CDF of a variable XX, find the CDF of a related variable YY, and hence its PDF, e.g. where Y=X3Y = X^3)

📊 The Probability Density Function (PDF)

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What is a Continuous Random Variable?

A continuous random variable is a random variable that can take all values in an interval. Unlike discrete random variables, continuous random variables are used to model quantities we measure, such as time, length, weight, or temperature.

🔑 Key Concept: For a continuous random variable, the probability that it equals any specific value is always zero: P(X=a)=0P(X = a) = 0. This is because there are infinitely many possible values in any interval.

Definition of a Probability Density Function

A probability density function (PDF) describes the probability distribution of a continuous random variable. Instead of using a table (as with discrete variables), we use a function f(x)f(x).

Conditions for a Valid PDF

For a function f(x)f(x) to represent a probability density function, it must satisfy two essential conditions:

Condition 1: Non-negativity

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

The probability density function can never be negative since probability cannot be negative.

Condition 2: Total Probability

∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1

The total area under the PDF curve must equal 1, representing the total probability.

Important: Both conditions must be satisfied for f(x)f(x) to be a valid probability density function.

Finding Probabilities Using the PDF

To find the probability that XX falls between two values aa and bb, we integrate the PDF between those values:

P(a<X<b)=∫abf(x) dxP(a < X < b) = \int_{a}^{b} f(x) \, dx

Since P(X=a)=0P(X = a) = 0 for continuous random variables, we have:

P(X<a)=P(X≤a)andP(X>a)=P(X≥a)P(X < a) = P(X \le a) \quad \text{and} \quad P(X > a) = P(X \ge a)

Piecewise Probability Density Functions

Sometimes, probability density functions are represented by different functions over different parts of the domain. These are called piecewise functions.

Example: A piecewise PDF might be defined as:

f(x)={k(x+1)1≤x<4k4≤x≤80otherwisef(x) = \begin{cases} k(x+1) & 1 \le x < 4 \\ k & 4 \le x \le 8 \\ 0 & \text{otherwise} \end{cases}

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