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From the way you write the information, it seems that you assume you have only one parameter to estimate ($\theta$) and you consider one random variable (the observation $X$ from the sample). This makes the argument much simpler so I will carry it in this way. You use the information when you want to conduct inference by maximizing the log likelihood. That log-likelihood is a function of $\theta$
Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?
I'm reading a book about Haskell, a programming language, and I came across a construct defined "algebraic data type" that looks like data WeekDay = Mon | Tue | Wed | Thu | Fri | Sat | Sun That simply declares what are the possible values for the type WeekDay. My question is what is the meaning of algebraic data type (for a mathematician) and how that maps to the programming language construct?
Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time. Do you know of any other concepts like these?
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What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. Answers at any level of sophistication are welcome.
I came across the statement that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations. How exactly is Yoneda-lemma a generalization of Cayley`s theorem? Can Cayley's theorem be deduced from Yoneda lemma, is it a generalization of a particular case of Yoneda, or is this instead, a philosophical statement? To me, it seems that Y
Let's go through some questions in order and see where it takes us. [Or skip to the bit about complex numbers below if you can't be bothered.] What are natural numbers? It took quite some evolution, but humans are blessed by their ability to notice that there is a similarity between the situations of having three apples in your hand and having three eggs in your hand. Or, indeed, three twigs or th
Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality. Suppose that $U: D \to C$ is a functor from a category $D$ to a category $C$, and let $X$ be an object of $C$. A universal morphism from $X$ to $U$ [...] consists of a
Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. Online articles say that these methods are 'related' but never specify the exact relation. What is the intuitive relationship between PCA and SVD? As PCA uses the SVD in its calculation, clearly
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