Course: Introduction to Statistics (18-week generic edition)
Module: Week 6 of 18 · planned around two ~75-minute sessions
Objective covered: Objective 4 — probability rules and random variables (this week: the random-variables half).

This file holds two pieces: (A) the Module 6 Overview page ("Start Here") and (B) the Welcome Announcement that drips out when the module opens. All timing is relative — "start of Week 6," "end of Week 6" — and maps onto real dates when the adopting instructor sets the term calendar.

(A) Module 6 Overview — Start Here

Welcome to Week 6: Random Variables

This is your home base for the week. Read it first, then work the checklist below from top to bottom. Everything you need is linked inside the module.

Last week you learned to measure chance itself. This week, chance gets a price tag. A claw-machine play, an extended warranty, a raffle ticket, a board-game draw — each is a number that hasn't happened yet, and this week you'll learn to say exactly what such a number is worth (the expected value) and how wildly it swings (the standard deviation). This is the machinery hiding inside every insurance premium, every game of chance, and — starting next week — every statistical model in the rest of the course.

The week's big question

"When an outcome is uncertain but has numbers attached — a prize, a payout, a count — what is that uncertain number actually worth, and how wildly does it swing?"

By the end of the week you'll be able to take any uncertain number, write down its possible values with their chances, and answer the two questions that matter: what's typical — and how far off typical should I expect to be?

By the end of this week, you can…

Use this as a checklist. If you can do all five out loud, you're ready for the quiz.

• [ ] Recognize a random variable and tell a discrete one (you count it) from a continuous one (you measure it).
• [ ] Check a distribution's legitimacy — every probability between 0 and 1, total exactly 1 — and read probabilities like P(X ≥ 2) from the table.
• [ ] Compute and interpret E(X) = Σ x·P(x) as a long-run average — remembering it need not be a possible value ("expected value is what you'd average, not what you'd expect").
• [ ] Compute Var(X) and SD(X) by the weighted squared-deviation method — and report the SD, not the variance, as your answer.
• [ ] Predict what a + bX does: adding shifts the center only; multiplying stretches center and spread — and explain why, for a continuous variable, probability is area and P(X = exact value) = 0.

What to do this week, in order

The table below lists the week's items in working order, with what each is worth and when it's due.

Heads-up on the AI work: in this course the chatbot drafts, and you judge. This week's classic chatbot blunder: asked for an expected value, it averages the values and ignores the probabilities — for the claw machine in the chapter, that's answering 4 when the truth is $0.80. It also loves handing you a variance dressed up as an SD. Catching both is literally graded in the lab.

Late policy reminder: 10% off per day late. If life happens, reach out to your instructor before the deadline — early is always easier.

How to succeed this week

• Lead with the table. Every discrete problem starts the same way: values row, probabilities row, legitimacy check (sum = 1 — exactly). Only then compute. The check takes five seconds and catches half of all errors before they happen.
• Memorize two tiny hooks. "Expected value is what you'd average, not what you'd expect" — E(X) is a long-run average, not a prediction, and it doesn't have to be a possible value. And "adding shifts the center; multiplying stretches both."
• Say every answer in words. "E(X) = 1.4" is half an answer; "about 1.4 sell-outs per day, on average, over many days" is the whole one. The quiz and the assignment both grade the words.
• Report the SD, not the variance. Variance is scratch work in squared units. One square root turns it into the number you actually say out loud.
• For continuous variables, think area. No table exists; intervals carry the probability, and a single exact value carries none. If you remember one chant: no width, no area, no probability.

You built the chance toolkit in Week 5; this week attaches it to money, counts, and scores. Come to the first session ready to vote on whether you'd play a $1 claw machine.

(B) Welcome Announcement — Module 6

Release setting: drips at the start of Week 6 (offset = 0 days from module start) — not before. If your platform won't preserve the scheduled release on import, post it as a draft labeled "Release: start of Week 6."

Subject: Week 6 — what is a chance actually worth?

Hi everyone, and welcome to Week 6!

Last week you put numbers on chance — the probability a card, a roll, or a customer goes one way or the other. This week, chance gets a price tag. A $1 claw-machine play that might pay $10. A $30 protection plan that might cover a $120 replacement. A raffle ticket. This week's machinery — random variables and expected value — tells you what any of those deals is really worth, and it's the exact math insurance companies and arcades use to price them.

This week — Random Variables — we tackle the big question: when an uncertain outcome has numbers attached, what is it worth, and how wildly does it swing? By the end of the week you'll compute a deal's long-run value E(X), measure its wobble with the SD, and know what happens when fees are subtracted or rates multiplied.

Three things not to miss:
1. Chapter 6 is your primary reading — start there; the claw machine on page one carries the whole week.
2. Data Lab 6 has your spreadsheet run a raffle: you'll compute what a ticket is worth by formula, then simulate 1,000 draws and watch the long-run average walk right up to your answer. Due at the end of Week 6.
3. Discussion 6 — "The Warranty Question" — is an extended warranty ever the smart buy? Your initial post is due two days before the week ends, so classmates have time to reply.

One heads-up: the expected value has a famous twist — it's usually a value that can't actually happen on any single play. That's not a bug, and by mid-week you'll be explaining why to your chatbot (which will get it wrong).

Open the Start Here / Module Overview page first — it lays out everything in order with due points. Bring an opinion to the first session: would you play a $1 claw machine that pays out 80 cents?

See you in the course!