Expectation, Variance, and Statistical Reasoning Clinic

Retrieval Prompts

1. State the definition of expectation for a discrete random variable.
2. State linearity of expectation.
3. State the expectation of an indicator variable.
4. Define variance in words.
5. State what the law of large numbers says.
6. State what the central limit theorem adds beyond the law of large numbers.

Compare and Distinguish

Separate these pairs clearly:

• expectation versus most likely value
• variance versus probability of error
• covariance versus independence
• LLN versus CLT
• probability statement versus confidence statement

Common Mistake Check

For each statement, identify the error:

1. "Linearity of expectation requires independence."
2. "Since the expected value is 3.5, the next roll is likely to be 3.5 in effect."
3. "Zero covariance always implies independence."
4. "A 95% confidence interval means a 95% probability the fixed parameter is inside it."
5. "The LLN means a tail is due after many heads."

Mini Application

Do the setup for each problem:

1. use indicators to model the expected number of occupied hash buckets
2. explain why two strategies with equal expectation may still differ sharply in reliability
3. explain how averaging repeated noisy measurements changes spread
4. rewrite one sloppy confidence claim into precise language

Evidence Check

This page is complete only if you can explain what expectation and variance mean operationally, not just symbolically.