Investigation of the Background Radiation and Poisson Statistics - - PowerPoint PPT Presentation

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investigation of the background radiation and poisson

Investigation of the Background Radiation and Poisson Statistics - - PowerPoint PPT Presentation

Mar 05, 2023 466 likes •976 views

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

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SLIDE 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

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SLIDE 2

Random Variables

Question: What does it mean for something to be “random”?

Jason Gross (8.13) Poisson Statistics October 7, 2011 2 / 26

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Random Variables

XKCD’s Answer:

Jason Gross (8.13) Poisson Statistics October 7, 2011 3 / 26

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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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

f:

the sample space Ω — an arbitrary non-empty set, the σ-algebra F ⊆ 2Ω (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 Ai ∈ F for i = 1, 2, . . ., then also ( i Ai) ∈ F the probability measure P : F → [0, 1] — a function on such that: ◮ P is countably additive: if {Ai} ⊆ F is a countable collection of pairwise disjoint sets, then P( Ai) = P(Ai), 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

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SLIDE 10

Random Variables

Yikes!

Jason Gross (8.13) Poisson Statistics October 7, 2011 6 / 26

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SLIDE 11

Random Variables

Let’s try again.

Jason Gross (8.13) Poisson Statistics October 7, 2011 7 / 26

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SLIDE 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

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SLIDE 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

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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

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Random Variables

Question: Why do we care?

Jason Gross (8.13) Poisson Statistics October 7, 2011 8 / 26

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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

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SLIDE 17

Random Variables

Question: What does it mean for something to be “random”?

Jason Gross (8.13) Poisson Statistics October 7, 2011 9 / 26

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SLIDE 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

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SLIDE 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

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SLIDE 20

Random Variables

Law of Large Numbers: lim

n→∞ P

|Xn − µ| > ε
= 0

(weak law) P

lim

n→∞ Xn = µ

= 1 (strong

law)

Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

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Random Variables

Law of Large Numbers: lim

n→∞ P

|Xn − µ| > ε
= 0

(weak law) P

lim

n→∞ Xn = µ

= 1 (strong

law)

Jason Gross (8.13) Poisson Statistics October 7, 2011 10 / 26

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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

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SLIDE 23

Random Variables

Question: Why is the law of large numbers enough? Answer:

Jason Gross (8.13) Poisson Statistics October 7, 2011 11 / 26

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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

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SLIDE 25

Binomial Distribution

Consider an event with two

utcomes.

Jason Gross (8.13) Poisson Statistics October 7, 2011 12 / 26

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SLIDE 26

Binomial Distribution

P(x; n, p) = n x

pxqn−x

= n! x!(n − x)!pxqn−x

Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

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SLIDE 27

Binomial Distribution

P(x; n, p) = n x

pxqn−x

= n! x!(n − x)!pxqn−x

Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

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SLIDE 28

Binomial Distribution

P(x; n, p) = n x

pxqn−x

= n! x!(n − x)!pxqn−x

Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

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SLIDE 29

Binomial Distribution

P(x; n, p) = n x

pxqn−x

= n! x!(n − x)!pxqn−x

Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

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SLIDE 30

Binomial Distribution

P(x; n, p) = n x

pxqn−x

= n! x!(n − x)!pxqn−x

Jason Gross (8.13) Poisson Statistics October 7, 2011 13 / 26

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Binomial Distribution

n 10 p 0.3 2 4 6 8 10 x 0.05 0.10 0.15 0.20 0.25 Px

Px n x px 1 pnx

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Poisson Distribution

Consider the limit of p ≪ 1, with µ = np fixed. P(x; µ) = µx x! e−µ

Jason Gross (8.13) Poisson Statistics October 7, 2011 15 / 26

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Poisson Distribution

Consider the limit of p ≪ 1, with µ = np fixed. P(x; µ) = µx x! e−µ

Jason Gross (8.13) Poisson Statistics October 7, 2011 15 / 26

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Poisson Distribution

Μ 3 2 4 6 8 10 x 0.05 0.10 0.15 0.20 Px

Px Μx x Μ

Jason Gross (8.13) Poisson Statistics October 7, 2011 16 / 26

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Poisson Distribution

Question: Why do we care about this limit? Answer:

Jason Gross (8.13) Poisson Statistics October 7, 2011 17 / 26

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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

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SLIDE 37

Gaussian Distribution

Consider the limit for µ large. P(x; µ) = 1 µ √ 2π e−1

2

x−µ

µ

2

Jason Gross (8.13) Poisson Statistics October 7, 2011 18 / 26

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SLIDE 38

Gaussian Distribution

Consider the limit for µ large. P(x; µ) = 1 µ √ 2π e−1

2

x−µ

µ

2

Jason Gross (8.13) Poisson Statistics October 7, 2011 18 / 26

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Gaussian Distribution

Μ 100 Σ 10 80 100 120 140 x 0.01 0.02 0.03 0.04 Px

Px 1 2 Π Σ

1

2 xΜ Σ 2 Jason Gross (8.13) Poisson Statistics October 7, 2011 19 / 26

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Methodology

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

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SLIDE 41

Methodology

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

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Methodology

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

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Methodology

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

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Results

Number of Counts 1 2 3 4 5 6 10 20 30 40 50 Μ 1 s x 0.96 0.98 Χ2 PValue Curve Poisson 3.4 0.33 Normal 3.5102 5.51067 Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 21 / 26

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Results

Number of Counts 2 4 6 8 10 12 14 5 10 15 20 25 Μ 4 s x 5.5 2.3 Χ2 PValue Curve Poisson 6.0 0.65 Normal

84. 6.81013

Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 22 / 26

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Results

Number of Counts 5 10 15 20 5 10 15 20 Μ 10 s x 9.9 3.1 Χ2 PValue Curve Poisson 5.7 0.77 Normal

28. 0.0052

Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 23 / 26

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Results

Number of Counts 80 100 120 140 160 2 4 6 8 10 12 14 Μ 100 s x 115. 11. Χ2 PValue Curve Poisson 11. 0.55 Normal

10. 0.59

Frequency Bin Jason Gross (8.13) Poisson Statistics October 7, 2011 24 / 26

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Results

Poisson Normal µ χ2 P-value χ2 P-value µ ≈ 1 3.4 0.32 350 5.5 · 10−67 µ ≈ 4 6.0 0.65 84 6.8 · 10−13 µ ≈ 10 5.7 0.77 28 0.0052 µ ≈ 100 11 0.55 10. 0.59

Jason Gross (8.13) Poisson Statistics October 7, 2011 25 / 26

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SLIDE 49

Thank You! Any questions?

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