118 lines
4.0 KiB
TeX
118 lines
4.0 KiB
TeX
\documentclass{article}
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% set up telugu
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\usepackage{fontspec}
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\newfontfamily\telugufont{Potti Sreeramulu}[Script = Telugu]
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\usepackage{polyglossia}
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\setdefaultlanguage{english}
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\setotherlanguage{telugu}
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%other packages
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{physics}
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\usepackage{siunitx}
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\usepackage{todonotes}
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\usepackage[plain]{fancyref}
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\usepackage{luacode}
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\usepackage{titling}
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\usepackage{enumerate}
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% custom deepak packages
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% \usepackage{luatrivially}
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\usepackage{subtitling}
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\begin{luacode*}
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math.randomseed(31415926)
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\end{luacode*}
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\title{Problem 1.2}
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\subtitle{Probability distributions}
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\author{\begin{telugu}హృదయ్ దీపక్ మల్లుభొట్ల\end{telugu}}
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% want empty date
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\predate{}
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\date{}
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\postdate{}
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% !TeX spellcheck = en_GB
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\begin{document}
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\maketitle
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Three distributions:
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\begin{enumerate}[(i)]
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\item Uniform distribution, $\rho_{uniform}(x) = 1$
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\item Exponential distribution, $\rho_{exponential}(t) = e^{-\flatfrac{t}{\tau}}$
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\item Gaussian distribution, $\rho_{gaussian}(v) = \flatfrac{e^{\flatfrac{-v^2}{2\sigma^2}}}{\left( \sqrt{2\pi} \sigma \right)}$
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\end{enumerate}
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\begin{enumerate}[(a)]
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\item Likelihoods
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\begin{enumerate}[(i)]
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\item What is the probabliity that a random number uniform on $\left[0, 1\right)$ will lie between $x = 0.7$ and $x = 0.75$.
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\item That the waiting time for radioactive decay will be more than twice exponential decay time $\tau$?
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\item That your score will be above $2 \sigma$ above the mean?
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\end{enumerate}
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\item Normalization, mean and standard deviation.
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\begin{enumerate}[(I)]
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\item Show that these probability distributions are normalised.
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\item What is the mean $x_0$ of each distribution?
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\item The standard deviation $\sqrt{\int \dd{x} \left( x - x_0 \right)^2 \rho(x)}$?
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\end{enumerate}
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\item Sums of variables
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\begin{enumerate}
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\item Draw a graph of the probability distribution of the sum $x + y$ for two random variables drawn from a uniform distribution.
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\item Argue that in general the sum $z = x+y$ of two random variables with distributions $\rho_1(x)$ and $\rho_2(y)$ will have a distribution given by $\rho(z) = \int \rho_1(x) \rho_2(z-x)$.
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\end{enumerate}
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\item Multidimensional probability distributions
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Given $\rho(v_x, v_y, v_z)$ is the big expression in Sethna $\prod_{i=(x, y, z)} \sqrt{\frac{M}{2\pi k T}} \exp(\frac{-Mv_i^2}{2 kT})$.
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\begin{enumerate}
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\item Show that the mean kinetic energy is $\frac{kT}{2}$.
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\item Show that the probability that the speed is $\abs{v}$ is given by the distribution
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\begin{equation}
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\rho_{Maxwell}(v) = \sqrt{\frac{2}{\pi}}\frac{v^2}{\sigma^3} \exp(\frac{-v^2}{2 \sigma^2})
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\end{equation}
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\end{enumerate}
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\end{enumerate}
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\section{Solution} \label{sec:solution}
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\subsection{Likelihoods} \label{subsec:sola}
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These are quite simple, straightforward integrals, so avoiding the details here.
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\begin{enumerate}[(i)]
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\item $\frac12$
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\item $\frac{1}{e^2}$
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\item $\flatfrac{(1 - \erf(\sqrt{2}))}{2} \approx 0.023$.
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Sethna really seems to give this one to the solver.
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\end{enumerate}
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\subsection{Standard Deviations}
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Normalisation is quite simple.
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Means are
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\begin{enumerate}[(i)]
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\item $\frac12$
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\item $\tau$
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\item $0$
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\end{enumerate}
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Standard deviations are
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\begin{enumerate}[(i)]
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\item
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\begin{align}
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\sigma_{uniform} &= \sqrt{\int_0^1 \dd{x} \left( x - \frac12 \right)^2} \\
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&= \sqrt{\left. \frac13 \left( x - \frac12 \right)^3 \right|_0^1} \\
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&= \sqrt{\frac13 \left(\frac14 \right)} \\
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\sigma_{uniform} &= \frac{1}{\sqrt{12}}
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\end{align}
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\item
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\begin{align}
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\sigma_{exponential} &= \sqrt{\int_0^\infty \dd{t} \left(t - \tau \right)^2 e^{\flatfrac{-t}{\tau}}} \\
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&= \tau
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\end{align}
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\item
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\begin{align}
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\sigma_{Gaussian} &= \sqrt{\int_{-\infty}^\infty \dd{v} \left(v\right)^2 \flatfrac{e^{\flatfrac{-v^2}{2\sigma^2}}}{\left( \sqrt{2\pi} \sigma \right)} } \\
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&= \sigma
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\end{align}
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\end{enumerate}
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\newpage
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\listoftodos
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\end{document}
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