Lesson Plan

• Define a continuous random variable and its probability density function (pdf).
• Identify the key properties of a pdf and use integration to find probabilities.
• Explain the cumulative distribution function (CDF) and its relationship to the pdf.
• Calculate the expectation and variance of continuous distributions using integral formulas.
• Apply linearity and independence properties to simplify expectation and variance calculations for transformed variables.

• Projector or interactive whiteboard
• Printed worksheet with pdf sketches and example problems
• Graphing calculator or computer with spreadsheet software
• Handouts showing uniform, exponential, and normal pdf graphs
• Whiteboard and markers

Begin with a quick visual: ask students to imagine measuring the exact height of a person and why a single exact value has probability zero. Review the discrete probability mass function and basic integration concepts they have already mastered. State that by the end of the lesson they will be able to write a pdf, compute its mean and variance, and solve typical textbook examples.

1. Do‑now (5'): Short quiz on discrete pmf and basic integration.
2. Mini‑lecture (15'): Define continuous RV, pdf properties, CDF; illustrate with uniform, exponential, normal sketches.
3. Guided practice (20'): Work through the uniform $U(2,5)$ example on the board, calculating mean and variance step‑by‑step.
4. Pair activity (15'): Students complete worksheet problems for exponential and normal distributions; teacher circulates to check understanding.
5. Consolidation (10'): Whole‑class review of the summary checklist and flowchart; students fill in missing steps on their handouts.
6. Exit ticket (5'): Each student writes one correct step for finding variance of a given pdf.

Recap the five‑step procedure for handling continuous random variables: identify support, verify the pdf, compute $E[X]$, compute $E[X^2]$, then obtain variance. Collect exit tickets as a quick retrieval check. For homework, assign additional problems from the textbook covering transformed variables and ask students to prepare a short explanation of how linearity simplifies expectation calculations.