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Graduate practice chapters

1. Graduate · Chapter 101

   Conditional expectation by groups · First group baseline a=-1.5; 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=-1.5; Second group baseline b=-1.

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2. Graduate · Chapter 102

   Beta–Bernoulli updating · Prior shape α=3; Prior shape β=5

   Use the first six fixed observations: 1, 1, 0, 1, 0, 0. Find the posterior shape parameters and the predictive probability of success on the next trial. Givens: Prior shape α=3; Prior shape β=5.

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3. Graduate · Chapter 103

   Smoothing a weak derivative · Smoothing width ε=0.7; 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=0.7; Corner location b=0.

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4. Graduate · Chapter 104

   Heat semigroup modes · Higher-mode amplitude a=0.8; Higher frequency k=6

   Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=0.8; Higher frequency k=6.

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5. Graduate · Chapter 105

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

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6. Graduate · Chapter 106

   Critical norm scaling · Final spatial factor λ=3.75; Lebesgue exponent p=2.5

   Evaluate at the current scale λ=2.65. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.75; Lebesgue exponent p=2.5.

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7. Graduate · Chapter 107

   Rare events and convergence · Magnitude exponent a=2; Rarity exponent b=2.25

   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=2.25.

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8. Graduate · Chapter 108

   Conditional expectation by groups · First group baseline a=-1; 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=-1; Second group baseline b=-0.5.

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9. Graduate · Chapter 109

   Beta–Bernoulli updating · Prior shape α=4; Prior shape β=1

   Use the first six fixed observations: 1, 1, 0, 1, 0, 0. Find the posterior shape parameters and the predictive probability of success on the next trial. Givens: Prior shape α=4; Prior shape β=1.

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10. Graduate · Chapter 110

    Contraction estimates · Contraction slope q=0.6; 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=0.6; Constant b=0.5.

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11. Graduate · Chapter 111

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

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12. Graduate · Chapter 112

    Variational energy · Initial sine amplitude a=1.5; Deviation penalty b=0.5

    Use the remaining amplitude A=0.6, two fifths of the initial amplitude. Calculate E[u_A], its derivative with respect to A, and the minimum within this trial family. Givens: Initial sine amplitude a=1.5; Deviation penalty b=0.5.

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13. Graduate · Chapter 113

    Fourier incompressibility projection · Frequency component k₁=3; Frequency component k₂=1.5

    Use θ=6π/5 radians. Compute the projected coefficient Pₖv=v−k(k·v)/|k|², its dot product with k, and its norm. Givens: Frequency component k₁=3; Frequency component k₂=1.5.

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14. Graduate · Chapter 114

    Rare events and convergence · Magnitude exponent a=0.5; Rarity exponent b=1

    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=1.

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15. Graduate · Chapter 115

    Conditional expectation by groups · First group baseline a=-0.5; Second group baseline b=0

    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=0.

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16. Graduate · Chapter 116

    Contraction estimates · Contraction slope q=0.7; 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=0.7; Constant b=1.

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17. Graduate · Chapter 117

    Smoothing a weak derivative · Smoothing width ε=0.8; 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=0.8; Corner location b=0.5.

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18. Graduate · Chapter 118

    Heat semigroup modes · Higher-mode amplitude a=1.2; Higher frequency k=3

    Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=1.2; Higher frequency k=3.

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19. Graduate · Chapter 119

    Variational energy · Initial sine amplitude a=2; Deviation penalty b=1

    Use the remaining amplitude A=0.8, two fifths of the initial amplitude. Calculate E[u_A], its derivative with respect to A, and the minimum within this trial family. Givens: Initial sine amplitude a=2; Deviation penalty b=1.

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20. Graduate · Chapter 120

    Coercivity in a finite model · First diagonal entry a=0.2; Second diagonal entry b=0.4

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.2; Second diagonal entry b=0.4.

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