What is the PDF of the product of two exponentially distributed iid random variables?
Product of two random variables that has exponential distribution
zero matrix to the power of 0 Why $0^0=I$?   I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem  Thanks in advance. Exclude lstinputlisting I want to save paper when I print my LaTeX file. That's why I want to hide all my code, inserted wtih lstinputlisting{}  I already know how to exclude tables and images or math but the lstinputlisting{} is a problem.   Hope someone can help me  \usepackage{comment} \excludecomment{figure} \let\endfigure\relax \excludecomment{table} \let\endtable\relax \excludecomment{align*} \let\endalign\relax \excludecomment{lstlisting} \let\endlstlisting\relax   edit: added lstlisting Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution ... Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$.  GOAL: Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$.  I realize that $(n-1)S^2/ \sigma ^2$ is chi-square with $(n-1)$ degrees of freedom.  I am given that $$E(S) = E\left[\frac{\sigma}{\sqrt{n-1}} \left[\frac{(n-1)S^2}{\sigma^2}\right]^{1/2}\right] = \frac{\sigma}{\sqrt{n-1}} \int_0^\infty v^{1/2} \frac{1}{\Gamma[(n-1)/2] 2^{(n-1)/2}} v^{(n-1)/2} e^{-v/2} \, dv$$  Can anyone help me understand why they have $v^{1/2}$ and $v^{(n-1)/2}$ (combining them $v^{n/2}$ as I think it should be $v^{1/2}$ and $v^{\frac{(n-1)}{2}-1}$ (combining both terms give $v^{(n-2)/2}$ since the pdf for chi-square distribution is $$ f(v) = \frac{v^{(x/2)-1} e^{-v/2}}{2^{x/2}\Gamma(x/2)},$$ where $x$ is degree of freedom?  Could anyone please tell me which one is the correct answer with explanation?  Thank you How to fix very slow Ubuntu booting? How can I run an external program from C and parse its output? I've got a utility that outputs a list of files required by a game.  How can I run that utility within a C program and grab its output so I can act on it within the same program? UPDATE: Good call on the lack of information.  The utility spits out a series of strings, and this is supposed to be portable across Mac/Windows/Linux.  Please note, I'm looking for a programmatic way to execute the utility and retain its output (which goes to stdout). I've been struggling with this all day today. I imagine it's not very hard, but my algebra skills are terrible. So, how can I show that if $m&gt;n$ and $a$ is a positive integer, then $$a^{2^n}+1 \mid a^{2^m}-1.$$ I just can't get a coherent picture of what I'm supposed to do. If this too (see yesterday's question) boils down to the Unique Factorization Theorem, I would be very grateful for any intuition on how to think about this! Show by induction that for all natural numbers n>3  $(1+\frac{1}{n})^n&lt;n$  Let $(1+\frac{1}{n})^n&lt;n$ be true !   We show that $(1+\frac{1}{n+1})^{n+1})&lt;n+1$  $(1+\frac{1}{n})^n(1+\frac{1}{n})&lt;n(1+\frac{1}{n})$   $(1+\frac{1}{n})^{n+1}&lt;n+1$  So now we just need to show that   $(1+\frac{1}{n+1})^{n+1}&lt;(1+\frac{1}{n})^{n+1}$  So here , If I show that :   $(1+\frac{1}{n+1})&lt;(1+\frac{1}{n})$   Does that mean im done with exercise ? I often see output from TeX with the warning overfull hbox, badness 10000. What does this message mean? Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise convergent.    I found this one in my textbook marked as "Fun things to solve" and although it may be fun, it's kind of hard to do.   To be honest I already failed at the first hurdle. I couldn't even find a sequence of functions which uniformly converges to $0$.   Is there a certain way of dealing with this kind of problem? Because it seems kind of hard for me to come up with sequences without a mathematical of way of doing so (or maybe there is a mathematical way I just don't know yet). Is it fair to buy a return ticket I know I will never use just because it's cheaper? I am planning a trip to Asia and then back to Rio. The thing is that I am traveling on flexible dates, so I am thinking of buying a one way ticket to Hong Kong, and I was planning to do the same when I want to come back.  But what I noticed is that, for example, a one way ticket from Melbourne to Rio costs no less than $2700 dollars for the month of May.  When I do a search for a round-trip from Melbourne to Rio and back, I can find flights for $2200.  I know I will not be using that return to Melbourne, but I'd still prefer to save those $500.   Can there be any issue with buying a round-trip with a return that I will never use? What's a better way to define individual link colours for a url in a macro? I've got a document where I'd like two colours of links - cyan for links in headers that since it's brighter, and navy blue for links in the main body which is more subtle/less distracting. I already tried using the xcolor package and changing the text colour with textcolor, like \textcolor{cyan}{\href{url.com}{link name}}, but it doesn't override whatever link or url colour I have defined in hypersetup.   So this is what I've done:   I've defined the colour navy:   \definecolor{navy}{HTML}{2F729C}    And also imported the hyperref package and defined commands to set the global link/url colours to cyan / navy.   \usepackage{hyperref}                         \hypersetup{colorlinks=true,linkcolor=navy,urlcolor=navy}                        \newcommand{\navylinks}{\hypersetup{linkcolor=navy,urlcolor=navy}}               \newcommand{\bluelinks}{\hypersetup{linkcolor=cyan,urlcolor=cyan}}     Then I've defined another command specifically for the header links, where I turn on cyan links globally, include the header text and link, and then turn on navy links again:   \newcommand{\headerlink}[1]{\bluelinks\href{#2}{#1}\navylinks}   My issue with this is that I feel like there could be a better way to do it than to set the global link colour every time I use a link macro; is there a way to define the link colour just for one link, like one would do with \textcolor{...}, that would be able to override hypersetup? Problem with Winbind/Samba if G is generated by {a,b} and ab=ba, then prove G is Abelian if G is generated by {a,b} and ab=ba, then prove G is Abelian  All elements of G will be of the form $a^kb^j,\,\, k,j \in \mathbb{Z}^+$   so I need to get $a^kb^j = b^ja^k$ Limit of $n/\ln(n)$ without L'Hôpital's rule I am trying to calculate the following limit without L'Hôpital's rule:     $$\lim_{n \to \infty} \dfrac{n}{\ln(n)}$$   I tried every trick I know but nothing works. You don't have to prove it by definition. Can I remove inline event handlers using jQuery? Why is the year NOT displayed in the questions and comments dates? Why is it that the year is never displayed in the questions or comments? Is it because it is the current year? I, and maybe others, wonder. Why are PCA eigenvectors orthogonal and what is the relation to the PCA scores being uncorrelated? I'm reading up on PCA, and I'm understanding most of what's going on in terms of the derivation apart from the assumption that eigenvectors need to be orthogonal and how it relates to the projections (PCA scores) being uncorrelated? I have two explanations provided below which use a link between orthogonality and correlation but fail to really explain it: , .  In the second picture it says that the condition $a_{2}^{T}a_{1}=0$  is imposed to ensure that the projection $y_{2}=Xa_2$ will be uncorrelated with $y_{1}=Xa_1$. Can someone provide an example to show why orthogonal vectors ensure uncorrelated variables?  What would happen in PCA if I chose vectors which are not orthogonal; is this even possible? I have read elsewhere that orthogonality is just a by-product of the covariance matrix being symmetric which would suggest that it's not possible to have non pairwise orthogonal eigenvectors. However in the first picture in search of the most 'suitable' matrix it almost seems as we are choosing $p_{1},\ldots,p_{m}$ to be orthogonal to give us a more convenient matrix $\textbf{P}$ one which has nice properties.  I've read other posts on this topic but have been unsatisfied with incorporation of the intuition with uncorrelated variables. I really appreciate any help in understanding this confusion!! Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$ Is $$\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$$ uniformly convergent on $(0,+\infty)$? Sadly, Weierstrass' criterion nor Cauchy's condition don't seem to work here and I just ran out of ideas. How to clip an Openlayers OSM Background to a Vector Boundary? I am doing some forecasting in R using Rob Hyndman's forecast package. The paper belonging to the package can be found .  In the paper, after explaining the automatic forecasting algorithms, the authors implement the algorithms on the same data set. However, after estimating both an exponential smoothing and ARIMA model they make a statement I do no understand (on page 17):     Note that the information criteria are not comparable.    I thought that an advantage of using AIC for model selection is that we can compare AIC values from different models, as long as they are estimated using the same data set. Is this incorrect?  This matter is of particular interest to me, as I was planning on combining forecasts from different model classes (e.g. exponential smoothing and ARIMA) using so called Akaike weights (see Burnham and Anderson, 2002, for discussion on Akaike weights)  References   Burnham, K. P., &amp; Anderson, D. R. (2002). Model selection and multi-model inference: a practical information-theoretic approach. Springer Verlag.