Some topics in analysis of boolean functions
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This document summarizes key points from a lecture on computational complexity and algorithms. It covers Fourier analysis and its applications to social choice functions, Arrow's impossibility theorem, and the hypercontractive inequality. The lecture also discusses algorithmic gaps, where the optimal solution differs from what can be achieved in polynomial time.
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Statistics Assignment 1 HET551 – Design and Developm.docx
Statistics Assignment 1
HET551 – Design and Development Project 1
Michael Allwright
Haddon O’Neill
Tuesday, 24 May 2011
1 Normal Approximation to the Binomial Distribution
This section of the assignment shows how a normal curve can be used to approximate the binomial distribution. This
section of the assignment was completed using a MATLAB function (shown in Listings 1) which would generate and
save plots of the various Binomial Distributions after normalisation, and then calculate the errors between the standard
normal curve and the binomial distribution.
The plots in Figures 1 and 2 show the binomial distribution for various n trials with probability p = 1
3
and p = 1
2
respectively. These binomial plots have been normalised so that they can be compared with the standard normal
distribution.
From these plots it can be seen that once the binomial distribution has been normalised, the normal approximation is
a good approach to estimating the binomial distribution. To determine its accuracy, the data in Table 1 shows the
evaluation of qn = P(bn ≥ µn + 2σn) for both the normal curve and binomial distribution.
qn = P(bn ≥ µn + 2σn) Calculation Error
n N(0, 1) B(n, 1
2
) B(n, 1
3
) B(n, 1
2
) B(n, 1
3
)
1 0.0228 0.0000 0.0000 -0.02278 -0.02278
2 0.0228 0.0000 0.0000 -0.02278 -0.02278
3 0.0228 0.0000 0.0370 -0.02278 0.01426
4 0.0228 0.0000 0.0123 -0.02278 -0.01043
5 0.0228 0.0313 0.0453 0.00847 0.02249
10 0.0228 0.0107 0.0197 -0.01203 -0.00312
20 0.0228 0.0207 0.0376 -0.00208 0.01486
30 0.0228 0.0214 0.0188 -0.00139 -0.00398
40 0.0228 0.0192 0.0214 -0.00354 -0.00134
50 0.0228 0.0164 0.0222 -0.00636 -0.00059
100 0.0228 0.0176 0.0276 -0.00518 0.00479
Table 1: Calculating the error of the normal approximation to the binomial for various n and p
2 Analytical investigation of the Exponential Distribution
For this part of the assignment the density function shown in Equation 1 was given.
f(x) = λe−λx for x ≥ 0 and λ ≥ 0 (1)
Before any calculations were attempted, the area under graph was checked to show that
´∞
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that the total probability of all possible values was 1.
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To find the CDF of the given function, the function was integrated with 0 and x being the lower and upper bound
respectively. This derivation is shown in Equations 2 to 4.
CDF =
ˆ x
o
f(x) dx =
ˆ x
o
λe−λx dx (2)
2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.33)
P
ro
b
a
b
ili
ty
(a) n = 1
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 0.67)
P
ro
b
a
b
ili
ty
(b) n = 2
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes (shifted left by u = 1.00)
P
ro
b
a
b
ili
ty
(c) n = 3
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Successes.
Complex numbers
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Some topics in analysis of boolean functions
- 1. Ryan O’Donnell Carnegie Mellon University
- 9. ℝ 3 (+1,+1,+1) (−1,−1,−1) (+1,+1,−1) (+1,−1,+1) (−1,+1,+1) − 1 − 1 − 1 +1 +1 +1 +1 − 1
- 10. ℝ 3 (+1,+1,+1) (−1,−1,−1) +1 +1 +1 +1 − 1 − 1 − 1 − 1
- 11. ℝ 3 (+1,+1,+1) (−1,−1,−1) − 1 − 1 − 1 +1 − 1 − 1 − 1 − 1
- 12. ℝ 3 (+1,+1,+1) (−1,−1,−1) +1 +1 +1 +1 +1 +1 +1 − 1
- 13. ℝ 3 (+1,+1,+1) (−1,−1,−1) +1 +1 +1 +1 +1 +1 +1 +1
- 14. ℝ 3 (+1,+1,+1) (−1,−1,−1) − 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1
- 15. ℝ 3 (+1,+1,+1) (−1,−1,−1) − 1 − 1 +1 +1 − 1 +1 +1 − 1
- 16. ℝ 3 (+1,+1,+1) (−1,−1,−1) − 1 +1 − 1 +1 +1 − 1 +1 − 1
- 17. ℝ 3 (+1,+1,+1) (−1,−1,−1) +1 − 1 − 1 +1 +1 +1 − 1 − 1
- 18. ℝ 3 (+1,+1,+1) (−1,−1,−1) − 1 − 1 − 1 +1 +1 +1 +1 − 1
- 19. (+1,+1,+1) +1 +1 +1 +1 − 1 − 1 − 1 − 1 +1 +1 +1 +1 − 1 − 1 − 1 − 1 (+1,+1,−1) (+1,−1,−1)
- 20. =
- 22. =
- 25. ⇓ Rest: 0
- 29. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 30. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 31. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 32. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 33. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 34. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3}
- 39. {2} {1} ∅ {3} {1,3} {1,2} {2,3} {1,2,3} Maj(x 1 , x 2 , x 3 )
- 41. Inf i (f) = Pr [ f(x) ≠ f(x (⊕i) ) ] +1 +1 +1 +1 − 1 − 1 − 1 − 1
- 42. Inf i (f) = Pr [ f(x) ≠ f(x (⊕i) ) ] +1 +1 +1 +1 − 1 − 1 − 1 − 1
- 44. LMN Theorem: If f is in AC 0 then avg Inf i (f)
- 45. ⇒ avg Inf i (Parity n ) = 1 ⇒ Parity ∉ AC 0 ⇒ avg Inf i (Maj n ) = ⇒ Majority ∉ AC 0
- 50. {2} {1} ∅ {3} {1,3} {1,2} {2,3} [3]
- 61. • • • • • • “ C > A > B ” “ A > B > C ” “ B > C > A ” A > B? B > C? C > A? f( ) f( ) f( ) = + = + = − + − + + + − + + − + − − − + − − − + − + − + − + − + +
- 62. • • • • • • “ C > A > B ” “ A > B > C ” “ B > C > A ” A > B? B > C? C > A? f( ) f( ) f( ) = + = + = − + − + + + − + + − + − − − + − − − + − + − + − + − + +
- 71. all use “Hypercontractive Inequality”
- 80. all use Hypercontractive Inequality
- 81. just use “q = 4” Hypercontractive Inequality
- 84. Opt best poly-time guarantee ln(N) “ Set-Cover is NP-hard to approximate to factor ln(N)”
- 85. Opt LP-Rand-Rounding guarantee ln(N) “ Factor ln(N) Algorithmic Gap for LP-Rand-Rounding”
- 86. Opt( S ) LP-Rand-Rounding( S ) ln(N) “ Algorithmic Gap Instance S for LP-Rand-Rounding”
- 90. Opt = 1/n
- 91. Opt = 1/n
- 92. Opt = 1/n
- 93. Opt = 1/n f(x) = sgn( )
- 94. Opt = 1/n f(x) = sgn(r 1 x 1 + ••• + r n x n ) ARV gets
- 95. Opt = 1/n ARV gets gap:
Editor's Notes
- Thank you for the kind introduction. It’s a pleasure to be here. Thanks also to Cynthia and the rest of the STOC PC for inviting me to give this talk.