Statistics · Unit 4: Probability, Random Variables, and Probability Distributions · 14 min read · Updated 2026-05-11
Continuous Random Variables — AP Statistics
AP Statistics · Unit 4: Probability, Random Variables, and Probability Distributions · 14 min read
1. What is a Continuous Random Variable?★★☆☆☆⏱ 2 min
Unlike discrete random variables that only take distinct, countable values, a continuous random variable can take any numerical value within one or more intervals of real numbers. Common real-world examples include measurement of height, time to complete a task, or volume of liquid in a container.
This topic makes up roughly 3.5-5% of your total AP Statistics exam score, and acts as a foundation for normal distributions and inference questions that appear in both MCQ and FRQ sections.
2. Probability Density Functions (PDFs)★★☆☆☆⏱ 3 min
For continuous random variables, probability is measured as the area under a curve called the probability density function (PDF), denoted $f(x)$.
$f(x) \geq 0$ for all $x$ in the domain of $X$
The total area under $f(x)$ over the entire domain equals 1:
\int_{-\infty}^{\infty} f(x) dx = 1
To find the probability that $X$ falls between $a$ and $b$, calculate the area under $f(x)$ between those bounds:
P(a \leq X \leq b) = \int_{a}^{b} f(x) dx
Because $P(X = c) = 0$ for any single value $c$, the inequality signs do not change the result: $P(a \leq X \leq b) = P(a < X < b)$.
Exam tip: When asked to verify a PDF is valid, always explicitly check both conditions (non-negativity and total area = 1). AP graders will deduct points if you only check the total area condition.
3. Cumulative Distribution Functions (CDFs)★★★☆☆⏱ 4 min
The cumulative distribution function (CDF) of a continuous random variable $X$, denoted $F(x)$, gives the probability that $X$ is less than or equal to a specific value $x$.
The CDF simplifies probability calculations: for any $a < b$, $P(a \leq X \leq b) = F(b) - F(a)$. You can also recover the PDF from the CDF by differentiation: $f(x) = F'(x)$ at all points where $F(x)$ is differentiable.
Exam tip: Always write the CDF as a piecewise function covering all real $x$, not just the interval where the PDF is non-zero. AP graders require the full domain for full credit.
4. Expected Value and Variance of Continuous Random Variables★★★☆☆⏱ 4 min
The formulas for expected value and variance of continuous random variables are analogous to the discrete case, with integrals replacing sums over discrete outcomes.
Linearity of expectation holds for continuous random variables just as it does for discrete: for any constants $a$ and $b$, $E(aX + b) = aE(X) + b$.
Exam tip: Memorize the phrase 'expected of X squared minus square of expected X' to avoid mixing up the order of terms in the variance shortcut. A negative variance is impossible, so if you get a negative value, you flipped the terms.
5. The Uniform Continuous Distribution★★☆☆☆⏱ 3 min
The uniform continuous distribution is the simplest continuous distribution, where all intervals of the same length within the domain are equally likely.
Exam tip: Don’t waste time integrating for uniform distribution probabilities. Use the rectangle area shortcut to save time and avoid integration errors on the exam.
Common Pitfalls
Why: Students confuse continuous probability density functions with discrete probability mass functions, where the output is a probability.
Why: Students carry over this discrete random variable habit to continuous random variables.
Why: Students confuse discrete and continuous versions of the uniform distribution.
Why: Students mix up the order of terms in the shortcut formula when rushing on the exam.
Why: Students focus on the interval where the PDF is non-zero and forget the CDF must be defined for all real numbers.