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MathidiaMathematics Academy
Probability II
Lesson 31 of 75
Introduction

Continuous random variables

Objective. Use densities, distribution functions and transformations.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Continuous random variables take values over an interval and are described by a probability density function f(x) with P(a ≤ X ≤ b) = ∫a^b f(x)dx, rather than individual point probabilities. The cumulative distribution function F(x) = P(X ≤ x) satisfies F′(x) = f(x), and transformations of variables use the change-of-variables formula f_Y(y) = f_X(g⁻¹(y))|d/dy g⁻¹(y)| for a monotonic transformation Y = g(X). For X uniform on [0,1], Y = −ln(X) has density f_Y(y) = e^(−y) for y > 0, an exponential distribution.

Worked example

A continuous random variable X has density f(x) = 2x for 0 ≤ x ≤ 1. Find P(X ≤ 1/2).

1/2
Fraction bar showing 1 out of 2 parts shaded
  1. 1Set up the integral P(X ≤ 1/2) = ∫0^(1/2) 2x dx
  2. 2Antiderivative of 2x is x²
  3. 3Evaluate from 0 to 1/2: (1/2)² − 0²
  4. 4Simplify the result

Answer: 1/4

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Common mistake. A common error is treating f(x) as a probability itself, claiming P(X = x) = f(x), when in fact any single point has probability zero for a continuous variable. Probabilities come only from integrating the density over an interval.

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