User:EGM6341.s11.team1.Chiu/Mtg24

Mtg 24: Fri, 25 Feb 11

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More on Cauchy distribution (probability density function [[media: Nm1.s11.mtg23.djvu| page23-2]])

Application: Single-slit diffraction, Max Born (Born-Cauchy rule): Probability interpretation of light


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$$ \color{blue}\theta = \ $$ random variant with uniform probability density function, and varies from $$ -\frac{\pi}{2} \ $$ to $$ \frac{\pi}{2} \ $$.
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$$ \color{blue}f_{\theta} (\theta) = \ $$ probability density function for $$ \theta \ $$
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[[media: Nm1.s11.mtg24.djvu| page24-2]]


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$$ \color{blue}x= \ $$ random variant with probability density function $$f_{x}(x) \ $$ to be determined from $$f_{\theta}(\theta) \ $$
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Probability for    $$ \displaystyle \ \theta < \Theta = \mathbb{P} (\theta < \Theta)= \int^{\Theta}_{- \infty} f_{\theta}(t) dt $$ (1)
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Probability for    $$ \displaystyle \ x < X = \mathbb{P} (x < X)= \int^{X}_{- \infty} f_{x}(t) dt $$ (2)
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$$ \mathbb{P} (\theta < \Theta) \ \overset{=} \  \mathbb{P} (x < X) \ $$
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$$ x \ \overset{=} \ d \tan \theta \ $$ and $$ X \ \overset{=} \ d \tan \theta \ $$
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$$ \Rightarrow dx=d(1+ \tan^{2} \theta)d \theta \ $$
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$$ { \color{red}(3)} \Rightarrow \int^{\Theta}_{- \infty}f_{\theta}(\theta)d \theta \ = \ \int^{X}_{- \infty}f_{x}(x)dx \ $$
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$$ \Rightarrow \underbrace{f_{\theta}(\theta)}_{{ \color{blue} \frac{1}{\pi}} \ { \color{red} constant}}d \theta \ = \ f_{x}(x)\underbrace{dx}_{ \color{blue} d(1+\tan^{2}\theta)d\theta} \ $$
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$$ \Rightarrow f_{x}(x)= \frac{1}{\pi d(1+ \underbrace{\tan^{2}\theta}_{ \color{blue} ( \frac{x}{d})^{2}} }) \ $$
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[[media: Nm1.s11.mtg24.djvu| page24-3]]


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$$  \displaystyle f_{x}(x)= \frac{d}{\pi(d^{2}+x^{2})} $$     Cauchy Distribution
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Reference: Lemons, Introdution to stochastic processes in physics, 2002, p.26, p.29. Simulation: AI Access site
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[[media: Nm1.s11.mtg24.djvu| page24-4]] The Mean jumps around, and takes on large(extreme) values
 * nm1.s11.Mtg24.pg4.fig1.svg

[[media: Nm1.s11.mtg24.djvu| page24-5]] Even though histogram of events converges to Cauchy Distribution, the Mean continues to jump around, and takes on extreme values (heavy tails)
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