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Investigation of the Background Radiation and Poisson Statistics Jason Gross MIT - Department of Physics Jason Gross (8.13) Poisson Statistics October 7, 2011 1 / 26 Random Variables Question: What does it mean for something to be

  1. Investigation of the Background γ Radiation and Poisson Statistics Jason Gross MIT - Department of Physics Jason Gross (8.13) Poisson Statistics October 7, 2011 1 / 26

  2. Random Variables Question: What does it mean for something to be “random”? Jason Gross (8.13) Poisson Statistics October 7, 2011 2 / 26

  3. Random Variables XKCD’s Answer: Jason Gross (8.13) Poisson Statistics October 7, 2011 3 / 26

  4. Random Variables Wikipedia’s Answer: A random variable is a function from a probability space to a measurable space, typically the real numbers. Jason Gross (8.13) Poisson Statistics October 7, 2011 4 / 26

  5. Random Variables Wikipedia’s Answer: A random variable is a function from a probability space to a measurable space, typically the real numbers. Jason Gross (8.13) Poisson Statistics October 7, 2011 4 / 26

  6. Random Variables Wikipedia’s Answer: A random variable is a function from a probability space to a measurable space, typically the real numbers. Jason Gross (8.13) Poisson Statistics October 7, 2011 4 / 26

  7. Random Variables Wikipedia’s Answer: A random variable is a function from a probability space to a measurable space, typically the real numbers. Jason Gross (8.13) Poisson Statistics October 7, 2011 4 / 26

  8. Random Variables Wikipedia’s Answer: A random variable is a function from a probability space to a measurable space, typically the real numbers. Jason Gross (8.13) Poisson Statistics October 7, 2011 4 / 26

  9. Wikipedia’s Answer: “Let (Ω , F , P ) be a probability space and ( E , E ) a measurable space. Then an ( E , E ) -valued random variable is a function X : Ω → E which is ( F , E ) -measurable. “The expanded definition is following: a probability space is a triple consisting of: the sample space Ω — an arbitrary non-empty set, the σ -algebra F ⊆ 2 Ω (also called σ -field) — a set of subsets of Ω , called events, such that: ◮ F contains the empty set: ∅ ∈ F , ◮ F is closed under complements: if A ∈ F , then also (Ω \ A ) ∈ F , ◮ F is closed under countable unions: if A i ∈ F for i = 1 , 2 , . . . , then also ( � i A i ) ∈ F the probability measure P : F → [ 0 , 1 ] — a function on such that: ◮ P is countably additive: if { A i } ⊆ F is a countable collection of pairwise disjoint sets, then P ( � A i ) = � P ( A i ) , where ‘ � ’ denotes the disjoint union, ◮ the measure of entire sample space is equal to one: P (Ω) = 1.” Jason Gross (8.13) Poisson Statistics October 7, 2011 5 / 26

  10. Random Variables Yikes! Jason Gross (8.13) Poisson Statistics October 7, 2011 6 / 26

  11. Random Variables Let’s try again. Jason Gross (8.13) Poisson Statistics October 7, 2011 7 / 26

  12. Random Variables Question: What does it mean for something to be “random”? Answer: Jason Gross (8.13) Poisson Statistics October 7, 2011 7 / 26

  13. Random Variables Question: What does it mean for something to be “random”? Answer: Jason Gross (8.13) Poisson Statistics October 7, 2011 7 / 26

  14. Random Variables Question: What does it mean for something to be “random”? Answer: Why do we care? Jason Gross (8.13) Poisson Statistics October 7, 2011 7 / 26

  15. Random Variables Question: Why do we care? Jason Gross (8.13) Poisson Statistics October 7, 2011 8 / 26

  16. Random Variables Question: Why do we care? Answer: We want to be able to talk quantitatively about the relationship between measurements and theory. Jason Gross (8.13) Poisson Statistics October 7, 2011 8 / 26

  17. Random Variables Question: What does it mean for something to be “random”? Jason Gross (8.13) Poisson Statistics October 7, 2011 9 / 26

  18. Random Variables Law of Large Numbers: The average of a series of identical statistically independent random variables almost always converges to the true mean. Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

  19. Random Variables Law of Large Numbers: The average of a series of identical statistically independent random variables almost always converges to the true mean. Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

  20. Random Variables Law of Large Numbers: � � | X n − µ | > ε n →∞ P lim = 0 (weak law) � � n →∞ X n = µ lim = 1 (strong P law) Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

  21. Random Variables Law of Large Numbers: � � | X n − µ | > ε n →∞ P lim = 0 (weak law) � � n →∞ X n = µ lim = 1 (strong P law) Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

  22. Random Variables Law of Large Numbers: The average of a series of identical statistically independent random variables almost always converges to the true mean. Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

  23. Random Variables Question: Why is the law of large numbers enough? Answer: Jason Gross (8.13) Poisson Statistics October 7, 2011 11 / 26

  24. Random Variables Question: Why is the law of large numbers enough? Answer: Because everything is a random variable! Jason Gross (8.13) Poisson Statistics October 7, 2011 11 / 26

  25. Binomial Distribution Consider an event with two outcomes. Jason Gross (8.13) Poisson Statistics October 7, 2011 12 / 26

  26. Binomial Distribution � n � p x q n − x P ( x ; n , p ) = x n ! x !( n − x )! p x q n − x = Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

  27. Binomial Distribution � n � p x q n − x P ( x ; n , p ) = x n ! x !( n − x )! p x q n − x = Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

  28. Binomial Distribution � n � p x q n − x P ( x ; n , p ) = x n ! x !( n − x )! p x q n − x = Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

  29. Binomial Distribution � n � p x q n − x P ( x ; n , p ) = x n ! x !( n − x )! p x q n − x = Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

  30. Binomial Distribution � n � p x q n − x P ( x ; n , p ) = x n ! x !( n − x )! p x q n − x = Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

  31. Binomial Distribution x � p x � 1 � p � n � x P � x � � � n P � x � n � 10 0.25 p � 0.3 0.20 0.15 0.10 0.05 x 2 4 6 8 10 Jason Gross (8.13) Poisson Statistics October 7, 2011 14 / 26

  32. Poisson Distribution Consider the limit of p ≪ 1, with µ = np fixed. P ( x ; µ ) = µ x x ! e − µ Jason Gross (8.13) Poisson Statistics October 7, 2011 15 / 26

  33. Poisson Distribution Consider the limit of p ≪ 1, with µ = np fixed. P ( x ; µ ) = µ x x ! e − µ Jason Gross (8.13) Poisson Statistics October 7, 2011 15 / 26

  34. Poisson Distribution Μ x � �Μ P � x � � x � P � x � Μ � 3 0.20 0.15 0.10 0.05 x 2 4 6 8 10 Jason Gross (8.13) Poisson Statistics October 7, 2011 16 / 26

  35. Poisson Distribution Question: Why do we care about this limit? Answer: Jason Gross (8.13) Poisson Statistics October 7, 2011 17 / 26

  36. Poisson Distribution Question: Why do we care about this limit? Answer: It’s everywhere! (when we count statistically independent events) Jason Gross (8.13) Poisson Statistics October 7, 2011 17 / 26

  37. Gaussian Distribution Consider the limit for µ large. � 2 � x − µ e − 1 1 2 √ P ( x ; µ ) = µ 2 π µ Jason Gross (8.13) Poisson Statistics October 7, 2011 18 / 26

  38. Gaussian Distribution Consider the limit for µ large. � 2 � x − µ e − 1 1 2 √ P ( x ; µ ) = µ 2 π µ Jason Gross (8.13) Poisson Statistics October 7, 2011 18 / 26

  39. Gaussian Distribution 1 1 x �Μ 2 2 � Σ � P � x � � � � 2 Π Σ P � x � 0.04 Μ � 100 0.03 Σ � 10 0.02 0.01 x 80 100 120 140 Jason Gross (8.13) Poisson Statistics October 7, 2011 19 / 26

  40. Methodology NaI Scintillator Photo Multiplier Tube Voltage Divider +HV Power Counter Amplifier Preamplifier Supply Jason Gross (8.13) Poisson Statistics October 7, 2011 20 / 26

  41. Methodology NaI Scintillator Photo Multiplier Tube Voltage Divider +HV Power Counter Amplifier Preamplifier Supply Jason Gross (8.13) Poisson Statistics October 7, 2011 20 / 26

  42. Methodology NaI ⑦ light Scintillator Photo Multiplier Tube Voltage Jason Gross (8.13) Poisson Statistics October 7, 2011 20 / 26

  43. Methodology NaI Scintillator ③ ⑥ e − Photo Multiplier Tube Voltage Jason Gross (8.13) Poisson Statistics October 7, 2011 20 / 26

  44. Results Μ � 1 � s x � 0.96 � 0.98 50 Χ 2 P � Value Curve Poisson 3.4 0.33 40 Number of Counts 3.5 � 10 2 5.5 � 10 � 67 Normal 30 20 10 0 1 2 3 4 5 6 Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 21 / 26

  45. Results Μ � 4 � s x � 5.5 � 2.3 25 Χ 2 P � Value Curve 20 Poisson 6.0 0.65 Number of Counts 84. 6.8 � 10 � 13 Normal 15 10 5 0 2 4 6 8 10 12 14 Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 22 / 26

  46. Results Μ � 10 � s x � 9.9 � 3.1 20 Χ 2 P � Value Curve Poisson 5.7 0.77 Number of Counts 15 Normal 28. 0.0052 10 5 0 5 10 15 20 Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 23 / 26

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