Week 6 — What Should You Expect? A Short Random-Variable Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 6 of the term — compute two expected values and two standard deviations from the dice and the calendar, say what each one means, and explain the idea to someone new to statistics.

This week you learned to treat a number that chance sets as a random variable, to find where its long run settles with the expected value, and to measure how far one result swings with the standard deviation. Last week you put a number on chance; this week chance sets a number. In this assignment you work four short problems on the dice and the calendar, interpret each one, and explain the ideas to someone who has never studied statistics. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles.

The task

You will work with three chance processes from this week. One fair die has the faces 1, 2, 3, 4, 5, and 6, each equally likely. The sum S of two fair dice has an expected value of 7 and a standard deviation of about 2.4152. And a month picked at random from a common year has 28, 30, or 31 days; the 12 months hold 365 days, and the number of days D has a standard deviation of about 0.8620 days.

Work these four problems, showing your arithmetic:

1. One fair die has the faces 1, 2, 3, 4, 5, and 6, and each face is equally likely. What is the expected value E(X) of the face?
2. The 12 months of a common year hold 365 days, so a month picked at random has an expected length of 365 divided by 12 days. Its length in hours is 24 times its length in days. What is its expected length in hours, E(H)?
3. The number of days D in a month picked at random from a common year has a standard deviation of about 0.8620 days. What is the standard deviation of the month's length in hours, 24 times D?
4. The sum S of two fair dice has a standard deviation of about 2.4152. The average of the two dice is S divided by 2. What is the standard deviation of that average?

Beside each answer, add one brief interpretation: what that value means for the chance process, and one use or limit of it. For problem 1, say what the expected value describes over many rolls. For problem 2, say whether any single month actually has that many hours. For problem 3, say what multiplying by 24 does to the spread. For problem 4, compare the spread of the average with the spread of one die, which is about 1.71.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Explain it to a non-expert

The volunteer coordinator at a community food bank has never studied statistics. They plan volunteer hours month by month and ask you two things: why the "average month" of 730 hours never actually shows up on the calendar, and how much a month's hours really vary. Write a 3–5 sentence message that answers both questions using your results from problems 2 and 3, gives a reason tied to the calendar, and names one limit — for example, that an expected value describes the long run rather than any one month, or that the swing in hours comes entirely from the different month lengths. Explain any technical term in everyday words. More than one way of explaining it can be good.

For a different, ungraded example, imagine a coffee-shop owner who reads that the shop's "expected daily customers" is a certain number and asks what that means. Model answer: "Expected here means the average over many days, not a promise for tomorrow. Some days will be busier and some quieter, and the standard deviation tells you how big that day-to-day swing usually is. So staff for the typical day, but keep someone on call for the busy ones. One limit: if something changes, like a new office opening nearby, last year's average may not describe next month."

Use that pattern for the food bank, with your own explanation and evidence.

Deliverables

Submit a single random-variable report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value means for the chance process and one use or limit of it;
• your message to the coordinator, explaining why the average month of 730 hours never shows up on the calendar and how much a month's hours vary, with a reason tied to the calendar and one limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch of each distribution with a short note. Every form is graded by the same scoring criteria. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a spreadsheet for the arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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