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Graduate practice chapters

1. Graduate · Chapter 461

   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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2. Graduate · Chapter 462

   Contraction estimates · Contraction slope q=0.3; Constant b=0.5

   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=0.5.

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3. Graduate · Chapter 463

   Smoothing a weak derivative · Smoothing width ε=0.15; 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=0.15; Corner location b=0.25.

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4. Graduate · Chapter 464

   Coercivity in a finite model · First diagonal entry a=0.6; Second diagonal entry b=1.4

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.6; Second diagonal entry b=1.4.

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5. Graduate · Chapter 465

   Critical norm scaling · Final spatial factor λ=0.75; Lebesgue exponent p=4

   Evaluate at the current scale λ=0.85. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.75; Lebesgue exponent p=4.

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6. Graduate · Chapter 466

   Rare events and convergence · Magnitude exponent a=2; 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=2; Rarity exponent b=1.25.

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7. Graduate · Chapter 467

   Conditional expectation by groups · First group baseline a=1.5; Second group baseline b=-1.5

   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=1.5; Second group baseline b=-1.5.

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8. Graduate · Chapter 468

   Contraction estimates · Contraction slope q=0.4; 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=0.4; Constant b=1.

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9. Graduate · Chapter 469

   Smoothing a weak derivative · Smoothing width ε=0.2; 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=0.2; Corner location b=0.5.

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10. Graduate · Chapter 470

    Coercivity in a finite model · First diagonal entry a=0.8; Second diagonal entry b=1.6

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.8; Second diagonal entry b=1.6.

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11. Graduate · Chapter 471

    Critical norm scaling · Final spatial factor λ=1.25; Lebesgue exponent p=5

    Evaluate at the current scale λ=1.15. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.25; Lebesgue exponent p=5.

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12. Graduate · Chapter 472

    Rare events and convergence · Magnitude exponent a=0.5; Rarity exponent b=2.75

    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=0.5; Rarity exponent b=2.75.

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13. Graduate · Chapter 473

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

    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=2; Second group baseline b=-1.

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14. Graduate · Chapter 474

    Contraction estimates · Contraction slope q=0.6; 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=0.6; Constant b=2.

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15. Graduate · Chapter 475

    Smoothing a weak derivative · Smoothing width ε=0.25; Corner location b=0.75

    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=0.25; Corner location b=0.75.

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16. Graduate · Chapter 476

    Coercivity in a finite model · First diagonal entry a=1; Second diagonal entry b=1.8

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1; Second diagonal entry b=1.8.

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17. Graduate · Chapter 477

    Critical norm scaling · Final spatial factor λ=1.5; Lebesgue exponent p=5.5

    Evaluate at the current scale λ=1.3. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.5; Lebesgue exponent p=5.5.

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18. Graduate · Chapter 478

    Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=3

    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=3.

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19. Graduate · Chapter 479

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

    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=2.5; Second group baseline b=-0.5.

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20. Graduate · Chapter 480

    Contraction estimates · Contraction slope q=0.7; 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=0.7; Constant b=-2.

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