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\begin{center}
  {\bf University of Kentucky, Physics 335 \\
    Homework \#5, Rev. A, due Monday, 2024-10-14 }
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\vspace{.3 in}

{\bf 1-d. Gaussian Moments}---In this homework we will learn the techniques for evaluating Gaussian intetegrals, and to use them to investigate the covariance of the joint Gaussian integral.

{\bf \quad a)} Let $N=\int_{-\infty}^{\infty} dz\, e^{-z^{2}/2}$ be the normalization of the standard 1-d Gaussian distribution.  While this distribution cannot be integrated analytically, the 2-d Gaussian can, because its area element (Jacobian) in polar coordinates allows for a $u$-substitution.  Show that it is equal to $N^{2}=\int_{-\infty}^{\infty} dz_{1}\int_{-\infty}^{\infty}dz_{2}\, e^{-\chi^{2}/2}$, where $\vec\chi=(z_{1},z_{2})$, and integrate in cylindrical coordinates $(\chi,\phi)$, where $z_{1}=\chi \cos\phi$ and $z_{2}=\chi\sin\phi$, to get $N$ and normalize the 1-d Gaussian.  [{\it bonus}: what is the normalization of a $\nu$-d Gaussian?]

{\bf \quad b)} Perform a change-of-variables on the above two integrals (over $z$ and $\chi$, respectively) to $x$, where $\alpha x^{2}=z^{2}/2=\chi^{2}/2$, to evaluate $I_{n}(\alpha)=\int_{0}^{\infty} dx\, x^{n} e^{-\alpha x^{2}}$ for $n=0,1$. Take the derivative with respect to $\alpha$ of both sides of the above integral indentities repeatedly to evaluate $I_{n}$ for $n=2,3,4$.

{\bf \quad c)} [{\it bonus}: Perform another change of variables to express each integral $I_{n}$ in terms of the \href{https://en.wikipedia.org/wiki/Gamma_function}{Gamma function} $\Gamma(\nu)=\int_{0}^{\infty}t^{\nu-1}e^{-t}dt$.  This shows the symmetry between even and odd $n$.  Using the same technique as in b), show that $\Gamma(\nu+1)=\nu\Gamma(\nu)$, $\Gamma(1)=1$ and $\Gamma(\frac12)=\sqrt\pi$.  Thus $\Gamma(n+1)=n!$ for $n\in\mathbb{N}$, and $\Gamma(\nu)$ is a generalization of the factorial to all real numbers.]

{\bf \quad d)} Transform the moments $I_{n}$ from integrals over $z$ to integrals over $x=\mu+\sigma z$ and show that $\mu$ and $\sigma$ are the {\it mean} and {\it standard deviation}, respectively of the 1-d Gaussian distribution $p_{G}(x;\mu,\sigma)$.


{\bf 1-d: Average of Uniform Deviates}---By the \href{https://en.wikipedia.org/wiki/Central_limit_theorem}{\it Central Limit Theorem} (CLT), the average $\bar x$ of $n$ samples from almost any random variable $X$ tends to the Gaussian distribution as $n\to\infty$.

{\bf \quad a)} Calculate the mean $\mu$ and standard deviation $\sigma$ of the {\it uniform} random variable $X$ with probability distribution $p_{u}(x)= 1$ when $0<x<1$ and 0 otherwise.

{\bf \quad b)} Histogram the distribution of 100,000 averages of $n=1$, 2, 3, 5, 10, and 100 uniform deviates each.  Calculate the mean $\langle \bar x \rangle$ and standard deviation $s_{\bar x}$ of each sample distribution, and compare with $\mu_{\bar x}=\mu$, $\sigma_{\bar x}=\sigma/\sqrt{n}$ from the parent distribution.  Plot the Gaussian distribution function $p_{G}(\bar x; \mu_{\bar x},\sigma_{\bar x})$ together with each histogram.

{\bf \quad c)} Calculate $p_{\chi}(u;1)$, the $\chi^{2}$ distribution with $\nu=1$ degree of freedom by using the {\it invariant} $p_{\chi}(u;1)du = 2 p_{G}(z)dz$, where $u=\chi^{2}=z^{2}$.  The factor of two comes from combining the left and right tails.

{\bf \quad d)} For the $\bar x$ data with $n=100$ from part b), which approaches the Gaussian distribution, histogram $u=z^{2}$, where $z=(\bar x-\mu)/(\sigma_{\mu})=(\bar x-\mu)/(\sigma/\sqrt{n})$, and plot it together with the $\chi^{2}$ distribution $p_{\chi}(u;1)$ for $\nu=1$ degree of freedom.


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