Expected Value and Variance of a Discrete Random Variable · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One Number for a Whole Distribution

Mathematics · Probability · ages 16-17
Name ______________________   Date ____________

  1. What does the variance of a distribution measure?

    • the value the variable is most likely to take on one go
    • how far the values spread out around the expected value
    • the largest value the variable is able to produce at all
    • the number of different values that appear in the table

  2. E(X) always has to be one of the values the variable can take.

    Circle one:   True   False

  3. The variance is worked out as E(X squared) minus the square of E(X).

    Circle one:   True   False

  4. A variable takes values 2 and 4 with probabilities 0.75 and 0.25. What is E(X)?

    Answer: ______________

  5. Why does a rare large prize pull the expected value up less than its size suggests?

    • because its contribution is the prize multiplied by a small probability
    • because large prizes are worth less than the sign claims they are
    • because the calculation caps how much any one single value is able to contribute
    • because rare prizes are left out of the expected value calculation

  6. A stall charges 2 pounds a go and the expected winnings are 1.40. What does that mean for a player over an afternoon?

    • they gain about 60p on every go they play at the stall
    • they break even, since 1.40 is close enough to 2 pounds
    • they lose about 60p a go in the long run, which is what the stall keeps
    • the result depends entirely on how lucky that particular player happens to be

  7. For the same X, calculate E(X squared).

    Answer: ______________

  8. A variable takes values 1, 2 and 3 with probabilities 0.5, 0.3 and 0.2. What is E(X squared)?

    Answer: ______________

  9. Sam computes the variance as [E(X)] squared minus E(X squared) and gets a negative number. What has happened?

    • the distribution must have had a negative value somewhere inside it
    • he has the two terms the wrong way round, and the right order never goes negative
    • a negative variance is perfectly allowed, and it simply means the spread runs downward
    • he should have taken the square root first, before doing the subtraction

  10. A stall's expected payout per go is 0.90 and it charges 1 pound. Dee says it should charge 90p to be fair, and the stall says it must charge more to pay for the marquee. Who is right?

    • the stall, since a fair game is one that covers its own running costs
    • Dee, and no stall at a fair is allowed to make a profit on a game
    • both: 90p is the fair stake, and charging above it is how the stall earns
    • neither of them, since a stake and an expected payout cannot be compared at all
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Answer key

For grown-ups. Fold this page away before handing over the rest.

One Number for a Whole Distribution W1-mt_PFdo0Nm3FP-s1

  1. how far the values spread out around the expected value · E(X) says where the distribution sits and the variance says how widely it is scattered around that point. The two answer different questions about the same table.
  2. False · It is an average across many goes, so it can land between the possible values. A spinner paying 0, 1 or 5 pounds can have an expected value of 0.80.
  3. True · Square each value, weight it by its probability and add for E(X squared), then take off the square of the mean.
  4. 2.5 · Weight and add: 2 × 0.75 + 4 × 0.25 = 1.5 + 1 = 2.5.
  5. because its contribution is the prize multiplied by a small probability · A prize of 100 at a probability of 0.01 contributes 1 to the total, the same as a prize of 2 at a probability of 0.5. Size and rarity trade off against each other exactly.
  6. they lose about 60p a go in the long run, which is what the stall keeps · The player hands over 2 and gets back 1.40 on average, so the gap of 0.60 flows to the stall on every go. Luck decides any single go and not the long run.
  7. 4.9 · E(X squared) = 1(0.2) + 4(0.5) + 9(0.3) = 0.2 + 2.0 + 2.7 = 4.9.
  8. 3.5 · Square each value first, then weight: 1 × 0.5 + 4 × 0.3 + 9 × 0.2 = 0.5 + 1.2 + 1.8 = 3.5.
  9. he has the two terms the wrong way round, and the right order never goes negative · Var(X) = E(X squared) minus the square of E(X), in that order. Swapping them flips the sign, and a real variance can never be negative, because it measures a spread.
  10. both: 90p is the fair stake, and charging above it is how the stall earns · Fair is a defined thing: stake equals expected winnings, which is 90p. Charging a pound is not unfair in the everyday sense, it is a 10p margin, and the honest move is to be clear about it.
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