📈 Continuous Random Variables

What is a Continuous Random Variable?

A continuous random variable takes any value in an interval. Unlike discrete random variables (which take specific values like 1, 2, 3), continuous variables can be any real number within a range.

Examples:

• Height of students (150 cm to 210 cm)
• Time to complete a test (0 to 180 minutes)
• Temperature (could be 20.5°C, 20.57°C, 20.571°C, ...)

Probability Density Functions (PDF)

Instead of listing probabilities at specific values, continuous distributions use a probability density function f(x).

Key Properties:

1. $f(x) \geq 0$ for all x (function is non-negative)
2. Total area under the curve = 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$
3. Probability is area: $P(a \leq X \leq b) = \int_a^b f(x) \, dx$
4. Point probability is zero: $P(X = c) = 0$ for any single value

Why Area Under Curve = Probability

For continuous variables, probability is determined by area, not by the height of the function at a point.

Graphically:

• The total area under the curve represents 100% probability (or 1.0)
• The area between two x-values represents the probability the variable falls in that range
• A curve that never touches the x-axis means those x-values have zero probability

Uniform Distribution

The uniform distribution on interval [a, b] has constant density.

$f(x) = \frac{1}{b - a} \text{ for } a \leq x \leq b$

Mean: $\mu = \frac{a + b}{2}$ (midpoint)

Standard Deviation: $\sigma = \frac{b - a}{\sqrt{12}}$

Probability: $P(c \leq X \leq d) = \frac{d - c}{b - a}$ (rectangular area)

Uniform Distribution Example

Buses arrive every 15 minutes. Your arrival time is uniformly distributed between buses. What is the probability you wait less than 5 minutes?

Given: X ~ Uniform(0, 15) where X = wait time in minutes

$P(0 \leq X \leq 5) = \frac{5 - 0}{15 - 0} = \frac{5}{15} = \frac{1}{3} \approx 0.333$

Normal Distribution as Continuous RV

The normal distribution (bell curve) is the most important continuous distribution in statistics.

Properties:

• Symmetric about the mean μ
• Determined by two parameters: μ (mean) and σ (standard deviation)
• Approximately 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ
• Used to model many natural phenomena (heights, test scores, errors)

Notation: $X \sim N(\mu, \sigma^2)$

Common Mistakes

1. Confusing PDF height with probability: The y-axis value is not probability; only area is
2. Thinking $P(X = c) \neq 0$: For continuous distributions, the probability of any exact value is always 0
3. Not recognizing when to use continuous vs discrete: Discrete has specific values; continuous has intervals

AP Exam Tip

When dealing with continuous variables on the AP exam, always think "area under the curve." If asked for a probability, you need to find an area (using tables, calculators, or normal approximation). Never calculate probability for a single point; always use an interval.