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Friesen and Shine (2019) wanted to determine whether male Australian cane toads have different testes sizes in different parts of the species' range (edge of the range vs. core of the range). As part of the study, they needed to quantify how big a toad's testes are relative to the toad's body size. They decided to perform a linear regression of total testes mass (in mg) against body mass (in g) and use the residual for each toad as a measure of the toad's relative testes size. The Coefficient Estimates table for their least-squares regression procedure is shown below.
$\begin{array}{|ccccc|}\hline \text{Term}& \text{Coefficient}& \text{Standart Error}& t-<!--\; -\; -->value& Pr>\mid <!--\; \mid \; -->t\mid <!--\; \mid \; -->\\ \text{(intercept)}& 19.192& 43.082& 0.44547& 0.65643\\ \text{body mass}& 3.0063& 0.36387& 8.262& 1.5733e^{-<!--\; -\; -->14}\\ \hline\end{array}$
One toad in the dataset had a body mass of 128g and a total tested mass of 163 mg. Compute the residual corresponding to this toad and write a sentence interpreting the residual.

Interpreting Normal Quantite Plots. In Exercises 5–8, examine the normal quantite plot and determine whether the sample data appear to be from a population with a normal distribution.
Ages of Presidents The normal quantile plot represents the ages of presidents of the United States at the times of their inaugurations. The data are from Data Set 15 “Presidents” in Appendix B.

Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?
My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars (r,θ) on the domain (r,θ)∈[1,∞)×[0,2π) subject to boundary conditions
u(1,θ)=f(θ),limr→∞u(r,θ)=0,
where f is a known function on [0,2π). It can be shown (e.g. via separation of variables) that the general solution for u that is bounded at infinity is given by
u(r,θ)=a0+∑n=1∞r−n(ancosnθ+bnsinnθ),
which we can match against the Fourier coefficients of f to find an and bn. However we see that the solution only strictly decays at infinity if a0=0. This is a restriction on the allowed boundary data f(θ) in order for the solution to exist.
So my question comes in two parts:
Is there a way to get around this? I've been reading a solution in which they write the constant a0 as a Fourier series expansion of a square wave defined on the domain [0,4π). However surely this throws up half-angles that cannot be matched with u.
Assuming the answer to the above question is no, can someone provide an argument as to why the constant seen at r=1 has to be the same as the constant at infinity? It seems to me that in the above formulation we have specified too much boundary data and therefore the solution doesn't exist. But why??Thanks in advance!

Is there a mathematical basis for the idea that this interpretation of confidence intervals is incorrect, or is it just frequentist philosophy?
Suppose the mean time it takes all workers in a particular city to get to work is estimated as 21. A 95% confident interval is calculated to be (18.3,23.7).According to this website, the following statement is incorrect:
There is a 95% chance that the mean time it takes all workers in this city to get to work is between 18.3 and 23.7 minutes.
Indeed, a lot websites echo a similar sentiment. This one, for example, says:
It is not quite correct to ask about the probability that the interval contains the population mean. It either does or it doesn't.
The meta-concept at work seems to be the idea that population parameters cannot be random, only the data we obtain about them can be random (related). This doesn't sit right with me, because I tend to think of probability as being fundamentally about our certainty that the world is a certain way. Also, if I understand correctly, there's really no mathematical basis for the notion that probabilities only apply to data and not parameters; in particular, this seems to be a manifestation of the frequentist/bayesianism debate.
Question. If the above comments are correct, then it would seem that the kinds of statements made on the aforementioned websites shouldn't be taken too seriously. To make a stronger claim, I'm under the impression that if an exam grader were to mark a student down for the aforementioned "incorrect" interpretation of confidence intervals, my impression is that this would be inappropriate (this hasn't happened to me; it's a hypothetical).
In any event, based on the underlying mathematics, are these fair comments I'm making, or is there something I'm missing?

Benefits of factoring matrix multiplication into two matrix multiplications?
Assuming we have a linear transformation for vectors $\in <!--\; \in \; -->{\mathbb{R}}^n$ to vectors $\in <!--\; \in \; -->{\mathbb{R}}^m$ denoted y=Wx and that the goal is to learn the values of W (or at least the values that produce minimum loss) from data.
My question is as follows:
Is there any benefit from splitting the linear transformation to two matrices and learning the factors instead of learning the original one, i.e., if W=AB, then learn A and B instead of learning W? Are there any mathematical properties of such factoring that make it useful?
Unfortunately, I am unable to find a mathematical term for this matrix factorization (if this is a proper term to use), so it would be great if someone can tell me what should I be reading about.
Also, the only benefit I could find is that if $\mathbf{A}\in <!--\; \in \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->q},\mathbf{B}\in <!--\; \in \; -->{\mathbb{R}}^{q\times <!--\; \times \; -->n}$ and q<n, the number of parameters needed would be (n+m)$\times <!--\; \times \; -->$q instead of n$\times <!--\; \times \; -->$m.

The mean is different from the mode and the median
I am reading a study in which the mean age of a population is 41 years. The mode and median are however both 35 years. I do not have access to the raw data. How I can interpret the difference between the mean and the mode/median?

Bayes' Theorem Application Question
I'm reading Networks by Mark Newman and I'm confused about one of his applications of Bayes' theorem. He begins with the following equation:
$P(A,x,y\mid <!--\; \mid \; -->\text{data})=\frac{P(\text{data}\mid <!--\; \mid \; -->A,x,y)P(A)P(x)P(y)}{P(\text{data})}$
He then assumes the prior probabilities P(x) and P(y) are uniform and therefore constants. He then states that we cannot assume the same for P(A) and introduces the following as its prior probability $P(A|p)$(where p is another probability on which A depends)
From this he updates the formula above as follows:
$P(A,x,y,p\mid <!--\; \mid \; -->\text{data})=\frac{P(\text{data}\mid <!--\; \mid \; -->A,x,y)P(A\mid <!--\; \mid \; -->p)P(p)P(x)P(y)}{P(\text{data})}$
Why doesn't P(data|A,x,y) include p (and why did he introduce p into the posterior probability)?

Uniform continuity of heat equation with $L^1$ data
I came across the following remark in my reading.
Remark. If the initial data is in $L^1$, then heat equation solutions approach 0 within the sup-norm. That is, if K(x,t) is the heat kernel and $f\in <!--\; \in \; -->L^1({\mathbb{R}}^n)$, then
$\underset{t\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{\int <!--\; \int \; -->}_{{\mathbb{R}}^n}K(x-<!--\; -\; -->y,t)f(y)dy=0,$
uniformly.
I can picture this working if f is uniformly continuous in ${\mathbb{R}}^n$ and $\underset{x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}f(x)=0$. Does $f\in <!--\; \in \; -->L^1({\mathbb{R}}^n)$ imply this somehow, and I'm not seeing this? I'm relatively new to $L^p$-spaces.
If so, then for $\mathrm{ϵ<!--\; ϵ\; -->}>0$, we can find some radius $r_{\mathrm{ϵ<!--\; ϵ\; -->}}>0$ such that |$|f(y)|<\mathrm{ϵ<!--\; ϵ\; -->}/2$.
Then we can bound the heat kernel, and use the fact that, $\int <!--\; \int \; -->K(x-<!--\; -\; -->y,t)dy=1$ to conclude, more or less.

I recently saw a proof that the real number field is not interpretable in the complex number field. But this required the axiom of choice, namely the existence of wild automorphisms of the complex numbers. Is there a way to prove it in ZF alone?

Shouldn't the probability of sampling a point from a continuous distribution be 0?
Hey I was reading about Gaussian EM algorithm in which you first calculate the likelihood of data points being sampled from a Gaussian and then adjust your mean and variance to maximize it. To calculate the probability of a point being sampled from a distribution we do
$P(x_i\mid <!--\; \mid \; -->\mathrm{\theta <!--\; \theta \; -->})=\frac{1}{\sqrt{2\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\sigma <!--\; \sigma \; -->}}^2}}\exp <!--\; \; -->\left(\frac{-<!--\; -\; -->(x_i-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}{)}^2}{2{\mathrm{\sigma <!--\; \sigma \; -->}}^2}\right)$
But how can you sample a point from a continuous distribution? Shouldn't this be zero? Moreover I read many times about sampling a point/data from some continuous distribution but I can't understand how could you do that as for continuous random variable X the $P(X=x_1)=0$, so how could yo sample data point(s) from a continuous distribution?
Or does this sampling have some other meaning. I've seen many questions on this platform but I couldn't get my answer.

What Implications Can be Drawn from a Binomial Distribution?
Hello everyone I understand how to calculate a binomial distribution or how to identify when it has occurred in a data set. My question is what does it imply when this type of distribution occurs?
Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?

I am reading about PCA and found an exercise that says
Show that when a N-dim set of data points X is projected onto the eigenvectors $V=[e_1{e}_2...e_n]$ of its covariance matrix $C=X{X}^T$, the covariance matrix of the projected data $C_p=Y{Y}^T$ is diagonal and hence that, in the space of the eigenvector decomposition, the distribution of X is uncorrelated.
What I have so far is
$Y=V^{T}X$
Therefore
$C_p=Y{Y}^T=V^{T}X(V^{T}X{)}^T=V^{T}X{X}^{T}V$
but there I got stuck. Any advise on how to proceed, moreover, what does "The covariance matrix of the projected data is diagonal" mean?

What does x% colder mean?
I see people using percentage increases to talk about temperature; for example
"Two weather predictions were presented with somewhat conflicting data, with the Energy Information Administration (EIA) predicting the 2007-08 winter to be 4 percent colder than 2006-07, but still 2 percent warmer than the 30-year average."
Is there any meaningful way to interpret this? I don't know what it means for one year to be 4 percent colder than another.

When is a Morphism between Curves a Galois Extension of Function Fields
It is known that the category of normal projective curves and dominant morphisms between them is equivalent to the opposite category of fields of transcendence degree 1 over C and algebraic extensions, so that birational invariants of curves are actually invariants of the function fields, etc.
In algebraic geometry, we have nice interpretations of what it means for an extension of function fields to be separable or purely inseparable (even if these ideas are difficult to visualize since they only occur in prime characteristic). Is there a similar geometric way to view Galois extensions? (Or, I suppose, normal extensions?). Does the fact that a Galois extension of degree n is a splitting field of a degree n polynomial have any geometric meaning?

Singular Vector Decomposition and PCA interpretation
The covariance matrix of any data X -> N * D (N samples and D dimension ) would be $Cov(X)=E[(X-<!--\; -\; -->E[X])(X-<!--\; -\; -->E[X]{)}^T]$. Let's Assume (X - E[X])=Y. Thus $Cov(X)=E[Y{Y}^T]$. Now from SVD we know that U and V are just eigenvectors of $A^{T}A$ and $A{A}^T$ respectively. Thus, we can just use Eigen decomposition of $Y{Y}^T$ as it is symmetric.
The confusion I face is how do you interpret this Cov(X) = Eigendecomposition $(Y{Y}^T)=PD{P}^{-<!--\; -\; -->1}$. From my point of view it is just a factorization of linear transformation.
How can we conclude that which dimension would have highest covariance from this factorization? I am having hard time interpreting it as anything but transformation, meaning if I multiply a vector with $PD{P}^{-<!--\; -\; -->1}$, the vector will get transformed to a space where eigen vectors are basis. In this case they are orthonormal and hence its just rotation and then scaled and then put back to original space.
All I can view it as a factorization which tells us about the Cov(X) as a transformation rather than Cov(X) itself.

Reparameterization of hyperprior distribution
I'm reading about Bayesian data analysis by Gelman et al. and I'm having big trouble interpreting the following part in the book (note, the rat tumor rate θ in the following text has: $\mathrm{\theta <!--\; \theta \; -->}\sim <!--\; \sim \; -->Beta(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})$:
Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution. Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}})=\log <!--\; \; -->(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\beta <!--\; \beta \; -->}})$ and $\log <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->})$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta$. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on a $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->})\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$, and so this particular prior density cannot be used here.
In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}},(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-<!--\; -\; -->1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,
$p(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})\propto <!--\; \propto \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-5/2},$
and on the natural transformed scale:
$p(\log <!--\; \; -->\left(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\beta <!--\; \beta \; -->}}\right),\log <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}))\propto <!--\; \propto \; -->\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{\beta <!--\; \beta \; -->}(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-5/2}.$
My problem is especially the bolded parts in the text.
Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}},(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-<!--\; -\; -->1/2})$"
Question (2): What is the appropriate Jacobian?
Question (3): How does the author arrive into the original and transformed scale priors?
To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.
P.S. if you need more information, or me to clarify my questions please let me know.

Generating Sets From Given Information
The problem I am working on is:
The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. [Hint: “Aor B” is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]
a.The next purchaser will request at least one of the three options.
b.The next purchaser will select none of the three options.
c. The next purchaser will request only an automatic transmission and not either of the other two options.
d.The next purchaser will select exactly one of these three options.
I am absolutely positive that $P(A)=40\mathrm{\%<!--\; \%\; -->}$, $P(B)=55\mathrm{\%<!--\; \%\; -->}$, $P(C)=70\mathrm{\%<!--\; \%\; -->}$, and $P(A\cup <!--\; \cup \; -->B\cup <!--\; \cup \; -->C)=85\mathrm{\%<!--\; \%\; -->}$. However, the pieces of data I am not quite certain about are $P(A\cup <!--\; \cup \; -->B\cap <!--\; \cap \; -->C^{\prime })=63\mathrm{\%<!--\; \%\; -->}$, $P(A\cup <!--\; \cup \; -->C\cap <!--\; \cap \; -->B^{\prime })=77\mathrm{\%<!--\; \%\; -->}$, $P(B\cup <!--\; \cup \; -->C\cap <!--\; \cap \; -->A^{\prime })=80\mathrm{\%<!--\; \%\; -->}$, do these values correspond to the rest of the data? If so, then I seems nearly impossible to be able to generate the Venn Diagram. Could someone help?
EDIT: What I am having a difficult time interpreting is, when they say in the question, "...63% request A or B." To me, that says only A or only B; and under this interpretation I would write $P(A\cup <!--\; \cup \; -->B\cap <!--\; \cap \; -->C^{\prime })=63\mathrm{\%<!--\; \%\; -->}$. Under André Nicolas' interpretation, "63% request A or B," means $P(A\cup <!--\; \cup \; -->B)=63\mathrm{\%<!--\; \%\; -->}$. If it is the case that André Nicolas is correct, then it seems like they should have stated in the question, "63% request A or B, A and C, B and C, or A and B and C."
Also, I solved the problem under André Nicolas' assumption, and for part d), I know the answer but I am sure how to put in it math symbols. How would I do that?

Upper bound for smallest eigenvalue of matrix $\Vert <!--\; \Vert \; -->\mathrm{\varphi <!--\; \varphi \; -->}(x){\Vert <!--\; \Vert \; -->}_2\le <!--\; \le \; -->1$
I am reading a paper which claims the following. But I am not sure how to show it rigorously. Any help is appreciated.
For all $x\in <!--\; \in \; -->\mathcal{X}$, assume the d-dimensional feature map is bounded such that $\Vert <!--\; \Vert \; -->\mathrm{\varphi <!--\; \varphi \; -->}(x){\Vert <!--\; \Vert \; -->}_2\le <!--\; \le \; -->1$. For any data distribution $\mathrm{\mu <!--\; \mu \; -->}$ consider the matrix
$A={\mathbb{E}}_{x\sim <!--\; \sim \; -->\mathrm{\mu <!--\; \mu \; -->}}[\mathrm{\varphi <!--\; \varphi \; -->}(x)\mathrm{\varphi <!--\; \varphi \; -->}(x{)}^{\mathrm{\top <!--\; \top \; -->}}]$
Prove that the largest possible minimum eigenvalue ${\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{min}}$ min of matrix A satisfies
${\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{min}}(A)\le <!--\; \le \; -->\frac{1}{d}$

I am reading "Problem Solving with Algorithms and Data Structures using Python" and the author is currently explaining the relation between comparisons and the Approximate Number of Items Left in an Ordered List.
I am struggling to perform the conversion of: $\left(\frac{n}{2^i}\right)=1$ to i=logn
I put this expression into MathWay and got back $i=lo{g}_2(1)$ and I'm a little confused on how these results are equivalent. I'm pretty rusty with logarithms so if you could help explain this conversion to me I'd greatly appreciate it.

I was reading about limits of data compression and I had this question that kept me thinking.
Alphabet set: Set of all the unique symbols in the input data
Q. If the distribution over the alphabet set is uniform then is it possible to compress data theoretically ? Standard compression methods (like Huffman codes) exploit the bias in the frequency distribution to obtain codes that have a lower average length of the input text. Is it possible to work without frequency?
Idea 1: One idea is that since the distribution is uniform over the symbols, then any assignment of codes would result in the same number of symbols in the range of the mapping, thus we would require the same number of bits to store that information. Hence the average length of compressed data should be greater than or equal to the original average length.
Idea 2: Another idea that pops up in my head is that if the distribution of symbols is uniform, why not create a new symbol set S' that is essentially a set of all the bi-grams of symbols. But, this distribution also needs to be uniform as the probability of a pair of symbols is same for all the possible cases.
These ideas force me to conclude that it is not possible to compress data if the distribution is uniform.
Please correct me if I am wrong at any point, also I would like to know your views on the problem.
Edit: If you can also provide a formal proof or a book for reference then it would be great.