Brownian Motion and Its Defining Properties · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

A random motion with normal steps

Mathematics · Probability · ages 23-24
Name ______________________   Date ____________

  1. A student says standard Brownian motion satisfies B_0 = 0. Is that right?

    Circle one:   True   False

  2. Standard Brownian motion starts at 0. What is Var(B_4)?

    Answer: ______________

  3. Standard Brownian motion starts at 0. What is the variance of B of 4?

    Answer: ______________

  4. What is the distribution of B of 4 in standard Brownian motion?

    • Normal with mean 0 and variance 4
    • Constant equal to 4
    • Uniform between -4 and 4

  5. What turns a scaled random walk into Brownian motion?

    • The Donsker limit of many small independent pieces
    • Rounding every step to the nearest integer
    • Freezing the walk after three steps

  6. By Brownian scaling, what is the distribution of B_9 / 3?

    • Normal with variance 9
    • Standard normal
    • Normal with mean 3
    • Uniform on [-3, 3]

  7. A student says Brownian increments over disjoint intervals are independent normals. Is that right?

    Circle one:   True   False

  8. For standard Brownian motion, what is E[(B_3 - B_1) squared]?

    Answer: ______________

  9. Kim models Brownian velocity as a smooth finite function of time. What is wrong?

    • She started the motion away from zero
    • She used normal instead of uniform steps
    • She assumed differentiability the paths never have

  10. Two students debate zooming into a Brownian path. Who is right?

    • Ari, who says it smooths out under magnification
    • Devi, who says jaggedness persists at every scale
    • Both, depending on the starting point
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Answer key

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

A random motion with normal steps W1-mt_T5KtWIk07Z-s1

  1. True · Pinning the start at the origin is the first defining property.
  2. 4 · Variance grows linearly with time: Var(B_4) = 4.
  3. 4 · Variance grows linearly with time.
  4. Normal with mean 0 and variance 4 · Fixed-time positions are normal with variance equal to time.
  5. The Donsker limit of many small independent pieces · Walks converge to Brownian paths under the right scaling.
  6. Standard normal · Scaling time by 9 scales values by 3, so dividing by 3 returns a standard normal.
  7. True · Independent normal increments are the second and third defining properties.
  8. 2 · The difference is normal with variance 3 - 1 = 2, and its square has mean 2.
  9. She assumed differentiability the paths never have · Nowhere differentiable means no smooth velocity exists.
  10. Devi, who says jaggedness persists at every scale · Scaling keeps the roughness at every zoom level.
Worksheet · LightMySky