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Graduate practice chapters

1. Graduate · Chapter 501

   Rare events and convergence · Magnitude exponent a=1.75; Rarity exponent b=1.25

   Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=1.75; Rarity exponent b=1.25.

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2. Graduate · Chapter 502

   Conditional expectation by groups · First group baseline a=-1; Second group baseline b=2

   Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-1; Second group baseline b=2.

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3. Graduate · Chapter 503

   Contraction estimates · Contraction slope q=-0.6; Constant b=1

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.6; Constant b=1.

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4. Graduate · Chapter 504

   Smoothing a weak derivative · Smoothing width ε=0.5; Corner location b=-0.25

   Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.5; Corner location b=-0.25.

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5. Graduate · Chapter 505

   Coercivity in a finite model · First diagonal entry a=2; Second diagonal entry b=2.8

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2; Second diagonal entry b=2.8.

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6. Graduate · Chapter 506

   Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=2.5

   Evaluate at the current scale λ=2.05. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.75; Lebesgue exponent p=2.5.

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7. Graduate · Chapter 507

   Rare events and convergence · Magnitude exponent a=2; Rarity exponent b=1.5

   Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=2; Rarity exponent b=1.5.

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8. Graduate · Chapter 508

   Conditional expectation by groups · First group baseline a=-0.5; Second group baseline b=2.5

   Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-0.5; Second group baseline b=2.5.

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9. Graduate · Chapter 509

   Contraction estimates · Contraction slope q=-0.5; Constant b=1.5

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.5; Constant b=1.5.

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10. Graduate · Chapter 510

    Smoothing a weak derivative · Smoothing width ε=0.55; Corner location b=0

    Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.55; Corner location b=0.

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11. Graduate · Chapter 511

    Coercivity in a finite model · First diagonal entry a=2.2; Second diagonal entry b=3

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.2; Second diagonal entry b=3.

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12. Graduate · Chapter 512

    Critical norm scaling · Final spatial factor λ=3; Lebesgue exponent p=3

    Evaluate at the current scale λ=2.2. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3; Lebesgue exponent p=3.

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13. Graduate · Chapter 513

    Contraction estimates · Contraction slope q=-0.4; Constant b=2

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.4; Constant b=2.

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14. Graduate · Chapter 514

    Smoothing a weak derivative · Smoothing width ε=0.6; Corner location b=0.25

    Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.6; Corner location b=0.25.

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15. Graduate · Chapter 515

    Coercivity in a finite model · First diagonal entry a=2.4; Second diagonal entry b=0.2

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.4; Second diagonal entry b=0.2.

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16. Graduate · Chapter 516

    Critical norm scaling · Final spatial factor λ=3.25; Lebesgue exponent p=3.5

    Evaluate at the current scale λ=2.35. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.25; Lebesgue exponent p=3.5.

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17. Graduate · Chapter 517

    Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=0.5

    Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=0.75; Rarity exponent b=0.5.

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18. Graduate · Chapter 518

    Conditional expectation by groups · First group baseline a=0.5; Second group baseline b=-2

    Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=0.5; Second group baseline b=-2.

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19. Graduate · Chapter 519

    Contraction estimates · Contraction slope q=-0.3; Constant b=-2

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.3; Constant b=-2.

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20. Graduate · Chapter 520

    Smoothing a weak derivative · Smoothing width ε=0.65; Corner location b=0.5

    Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.65; Corner location b=0.5.

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