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Indicator variables

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  1. Idea: Count with indicators

    Write a count as a sum of indicators, each if its event happens and if not.

    • The mean needs no independence
    • Pick the unit: seats, days, pairs, values

  2. Formula: Linearity of indicators

    The expected count is the sum of the chances that each unit is counted, dependent or not.

  3. Example: Sign flips in a report

    positions in random order, long and short. Expected number of neighbours with opposite signs?

    1. neighbouring pairs; mark each flip
    2. One pair flips with chance
    3. Expected flips:

  4. Example: Count the easy side

    orders go independently and uniformly to venues. Expected number of venues used?

    1. Mark the empty venues instead
    2. One is empty with chance : expect empty
    3. Used:

  5. Formula: The variance needs the pairs

    Linearity stops at the mean; for the spread, the covariance of each pair comes back.

  6. Idea: Indicator covariance stays simple

    is the indicator that both events happen.

    • Without replacement the covariance is negative

  7. Quick check: Spread of an audit

    Draw of orders without replacement; are flagged. Variance of the flagged count?

    Show the answer

    Answer:

    Each draw has variance , each pair covariance : .

  8. Common mistake: Linearity for the variance

    Adding up as if linearity covered the variance too. It works only when the covariances vanish.

    Instead The mean never cares about dependence; the variance always does. Count the pairs.

  9. Idea: Say it in the room

    Name the indicator in one sentence: let be if seat is counted.

    • Say that linearity needs no independence
    • Give one unit's chance, multiply, finish
    • Asked about spread: go to pairs and covariance signs