Continuous Random Variables and Density Functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Chance as area under a curve

Mathematics · Probability · ages 19-20
Name ______________________   Date ____________

  1. X is uniform on [0, 4]. Find P(1 < X < 3). Give your answer as a decimal.

    Answer: ______________

  2. A variable X is uniform on [0, 4], so its density is f(x) = 1/4 there and 0 elsewhere. Which check confirms f is a valid density?

    • The curve never dips below zero and the total area under it equals 1
    • The tallest point of the curve equals 1
    • The curve crosses the horizontal axis somewhere
    • The total area under the curve equals 0

  3. X is uniform on [0, 4], so its density is 1/4 there and 0 elsewhere. Which check confirms it is valid?

    • The curve never dips below zero and the total area equals 1
    • The tallest point of the curve equals 1
    • The total area under the curve equals 0

  4. Mia says P(X = 2) = 1/4 for X uniform on [0, 4], because f(2) = 1/4.

    Circle one:   True   False

  5. For a continuous variable, P(X = 5) = 0. Which conclusion about this fact is wrong?

    • An interval around 5 can still have positive chance
    • The value 5 is impossible and can never occur
    • The density height at 5 can still be positive

  6. For a continuous variable, P(X = 5) = 0. Which conclusion about this fact is wrong?

    • The value 5 is impossible and can never occur
    • Probabilities come from areas, not from single points
    • An interval around 5 can still have positive probability
    • The density at 5 can still be a positive number

  7. A variable has cumulative function F(x) = x/4 on [0, 4]. What is its density f(x) there?

    • 1/4
    • 4
    • x/2
    • 0

  8. The density f(x) = x over 8 on [0, 4] is valid. Find P(X < 2). Give your answer as a decimal.

    Answer: ______________

  9. Tom computes P(1 < X < 3) for uniform [0, 4] as f(2) = 1/4. What is wrong?

    • He used the height at a point instead of the area
    • He used the wrong interval endpoints
    • He should have divided by the width

  10. Buses arrive evenly between 0 and 20 minutes. What is P(the wait is between 5 and 10 minutes)? Give a decimal.

    Answer: ______________

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Answer key

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

Chance as area under a curve W1-mt_jlf4zDTeNm-s1

  1. 0.5 · The slice from 1 to 3 has width 2 and height 1/4, so the area is 0.5.
  2. The curve never dips below zero and the total area under it equals 1 · Height 1/4 is positive and the rectangle area is 4 times 1/4, which is 1, so both density rules hold.
  3. The curve never dips below zero and the total area equals 1 · Height 1/4 is positive and width 4 times 1/4 is 1.
  4. False · Density height is not chance; the chance at a point is zero.
  5. The value 5 is impossible and can never occur · Zero area at a point says nothing about nearby intervals.
  6. The value 5 is impossible and can never occur · Zero probability at a point does not mean impossibility. The value can occur; only intervals carry positive probability.
  7. 1/4 · Density is the derivative of the cumulative function, and the derivative of x/4 is 1/4.
  8. 0.25 · The triangle has base 2 and height 0.25, so half of 0.5.
  9. He used the height at a point instead of the area · Chance is width times height, not the height alone.
  10. 0.25 · Width 5 times height 1 over 20 gives 0.25.
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