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Real life uses of Moment generating functions In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in most courses the examples they provide for expectation and variance can be solved analytically using the definitions.   Are there any real life examples of distributions where finding the expectation and variance is hard to do analytically and so the use of m.g.f's was needed? I'm asking because I feel like I don't get to see exactly why they are important in the basic courses. What *is* an Artificial Neural Network? As we delve into  literature, we get to identify other methods with neuromorphic topologies ("Neural-Network"-like architectures). And I'm not talking about the . Examples are given below.  Then, it makes me wonder: what is the definition of an artificial Neural Network? Its topology appears to cover everything.    Examples:  One of the first identifications we make is between PCA and a linear Autoencoder with tied-weights in the encoder and decoder and thresholded activations in the bottleneck layer.  Also, a common identification is done between linear models (logistic regression in special) and a Neural Network with no hidden layer and a single output layer. This identification opens several doors.  . ? ANN. Gaussian Process? ANN (with single hidden layer with infinite hidden units).  And so, just as easily, we can incorporate arbitrary regularized versions with specialized loss functions of these algorithms into a Neural Network framework.  But the more we dig, the more similarities appear. I just stumbled into , which makes the identification of a specific ANN architecture with decision trees, allowing these to be learned by ANN methods (such as Gradient Descent backpropagation). From this we can construct Random Forests and Gradient Boosted Decision Trees from solely Neural Network topologies.  If everything can be expressed as an Artificial Neural Network, what defines an Artificial Neural Network? Is the term "errored out" a grammatically correct phrase, or just a colloquial one? How can I download this flash game for offline play? is one of the best flash games ever.    Is there some way to download it, so that I can play it offline? Methane’s absorption bands are centered at 3.2 and 7.2 microns -- far off the peak of the Planck spectrum for a 290 K blackbody near 16 microns.  Moreover, its absorption bands overlap with the water absorption region.  So why is methane considered a stronger greenhouse gas than carbon dioxide, which has an absorption band at 15 microns? Is there a difference between Therefor and Therefore? How to deal with factors with rare levels in cross-validation? Suppose in a regression analysis in R, I have a factor type independent variable with 3 levels in my train dataset. But in the test data set that same factor variable has 5 levels. Therefore I can not predict the response values for test dataset. What should be done in this case? How to put a variable in anchor tag Tool for straightening a bent metal bar of the gate? Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! This question is a possibly harder version of: .  Question.  Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, g(y)+f(x)\, f(y), \tag{2} \end{align} for all $x,y \in \mathbb{R} $. If $g$ is continuous at $x=0$ and not identically zero, then there exists an $\alpha\in\mathbb R$, such  $$ f(x)=\sin \alpha x\quad\text{and}\quad g(x)=\cos \alpha x. $$ Is there a pair of discontinuous $f$ and $g$ satisfying $(1)$ and $(2)$?  Update. If $\ell :\mathbb R\to\mathbb R$ is a linear functional over $\mathbb Q$ (i.e., $\ell(qx+ry)=q\ell(x)+r\ell(y)$, for all $x,y\in\mathbb R$ and $q,r\in\mathbb Q$), then $$ \sin\big(\ell(x)\big), \quad \cos\big(\ell(x)\big), $$ satisfy $(1)$ and $(2)$. Discontinuous such functionals do exist, and they are obtainable using Zorn's Lemma (equivalently the Axiom of Choice.) This takes care of the second question. About the complex nature of the wave function? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). Show that if terms of a series have ratio less than the ratio of convergent series, then the series converges We have two infinite series $\sum a_n $ and $\sum b_n$, such that $a_n &gt; 0,\ b_n &gt; 0$ and $a_{n + 1}/a_n \leq b_{n + 1}/b_n$ for all $n$. Show that if $\sum b_n$ converges, then $\sum a_n$ also converges.   Any hints on how to prove this? I can sort of picture why this must be true, but I'm not sure how to prove this rigorously. I'm not sure using the ratio test would help here.. Why do we need resistors in led How to make pdfsam work in (L)ubuntu 18.04? Massless charged particles Decrease available inventory item on Cognito Form, removing it when zero I want to limit inventory quantities on a sales form, so that it can start at say 100 and count backwards as applicants submit orders and then shut down that particular option when it reaches 0.    Is this possible with Cognito? I have two machines with RJ11 ports and I had a RJ11 Modular Telephone Cable lying around.I wondered if it was possible to create a network between the two machines with those RJ11 Modular Telephone Cable (back to back via RJ11 ports) much like a network we create between two computers with a single RJ45 Modular connector cable.  Is this setup possible at all ? If it is , what will be the maximum transfer speed i can achieve between these two computers ? Confidence interval for Bernoulli sampling I have a random sample of Bernoulli random variables $X_1 ... X_N$, where $X_i$ are i.i.d. r.v. and $P(X_i = 1) = p$, and $p$ is an unknown parameter.  Obviously, one can find an estimate for $p$: $\hat{p}:=(X_1+\dots+X_N)/N$.  My question is how can I build a confidence interval for $p$? Inductive proof of the closed formula for the Fibonacci sequence A question here on meta for instance may stick around for quite a while. A feature-request  maybe even for months. Then the status (tags) gets changed to  or  or whatever and the OP does not get notified.  On a normal post-edit we get a notification but retagging does not count for that. Please add retagging to the edit notifications.