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What does the third condition mean in the definition of a well-posed PDE (in the sense of Hadamard)?
A PDE is called well-posed (in the sense of Hadamard), if
1. A solution exists.
2. The solution is unique.
3. The solution should depend continuously on the initial and/or boundary data.
As far as I have understood, it means that the solution does not undergo a sudden deviation as we increment/decrement the initial/boundary data by a little amount.
Does it mean the same? How can I interpret this mathematically?

if a matrix is a transformation, then how data can be interpretated in terms of linear mapping?
We can interpret a matrix as a linear mapping in linear algebra using matrix representation. However, in machine learning or deep learning, we represent input data as a matrix. Then, how can this data be interpreted in terms of linear mapping?

Relating factors of a polynomial over GF(q) and $GF(q^m)$
I am reading Richard Blahut's book "Algebraic Codes for Data Transmission" and have come to an impasse in my understanding in section 5.3. In this section, the author wants to relate the factors of a polynomial over GF(q) and the factors of that same polynomial over a GF-extension field $GF(q^m)$.
There is a good example in the book that I will use here to ask my question. Specifically the GF fields in this example use q=2 and m=4, so there is the prime field GF(2) and the extension field $GF(2^4)$. The polynomial to be factored over both of these fields is one with particularly special properties, of the form $x^{q^m-<!--\; -\; -->1}-<!--\; -\; -->1$. So, the polynomial for this example is $x^{15}-<!--\; -\; -->1$.
The author first factors $x^{15}-<!--\; -\; -->1$ over GF(2). The prime factors are:
$x^{15}-<!--\; -\; -->1=(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$
This factorization over GF(2) makes sense to me and I can easily verify it in MATLAB.
The author then chooses two of the above prime factors and forms a new polynomial:
$$\begin{gathered} g(x)=(x^4+x^3+1)(x^4+x^3+x^2+x+1) \\ =x^8+x^4+x^2+x+1 \end{gathered}$$
The above multiplication over GF(2) also makes sense to me and I can easily verify it in MATLAB.
However, the author then changes to analyzing the same polynomial $g(x)=x^8+x^4+x^2+x+1$ in the extension field $GF(2^4)$. It seems to be implied that because $GF(2^4)$ in an extension field of GF(2), and because the polynomial in question is comprised of prime factors (over GF(2)) of the special composite polynomial x15−1, that this switch to the extension field $GF(2^4)$ is justified.
To analyze $g(x)$ in the extension field, we agree to represent the 16 elements of the extension field as the usual powers of the primitive field element alpha:
$0,1,\mathrm{\alpha <!--\; \alpha \; -->},{\mathrm{\alpha <!--\; \alpha \; -->}}^2,{\mathrm{\alpha <!--\; \alpha \; -->}}^3,{\mathrm{\alpha <!--\; \alpha \; -->}}^4,{\mathrm{\alpha <!--\; \alpha \; -->}}^5,{\mathrm{\alpha <!--\; \alpha \; -->}}^6,{\mathrm{\alpha <!--\; \alpha \; -->}}^7,{\mathrm{\alpha <!--\; \alpha \; -->}}^8,{\mathrm{\alpha <!--\; \alpha \; -->}}^9,{\mathrm{\alpha <!--\; \alpha \; -->}}^{10},{\mathrm{\alpha <!--\; \alpha \; -->}}^{11},{\mathrm{\alpha <!--\; \alpha \; -->}}^{12},{\mathrm{\alpha <!--\; \alpha \; -->}}^{13},{\mathrm{\alpha <!--\; \alpha \; -->}}^{14}$
The author claims that the aforementioned polynomial $g(x)=x^8+x^4+x^2+x+1$ has roots/zeroes over
$g({\mathrm{\alpha <!--\; \alpha \; -->}}^3)=0$
I am completely unable to make sense of this claim. It raises numerous questions in my mind, and the most basic of these questions is, how can I verify that $g(\mathrm{\alpha <!--\; \alpha \; -->})=0$ and that $g({\mathrm{\alpha <!--\; \alpha \; -->}}^3)=0$?
To show how I've tried to verify that $g(\mathrm{\alpha <!--\; \alpha \; -->})=0$ for $g(x)=x^8+x^4+x^2+x+1$, the author uses the following detailed representation of the elements of extension field $GF(2^4)$. Of course to describe the elements of any GF extension field, we must choose an irreducible polynomial for the reducing modulus, and throughout the book the author uses the modulus ${\mathrm{\alpha <!--\; \alpha \; -->}}^4+\mathrm{\alpha <!--\; \alpha \; -->}+1$, yielding the 16 field elements:
$$\begin{gathered} 0={\mathrm{\alpha <!--\; \alpha \; -->}}^{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}} \\ 1={\mathrm{\alpha <!--\; \alpha \; -->}}^0 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^1=\mathrm{\alpha <!--\; \alpha \; -->} \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^2={\mathrm{\alpha <!--\; \alpha \; -->}}^2 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^3={\mathrm{\alpha <!--\; \alpha \; -->}}^3 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^4=\mathrm{\alpha <!--\; \alpha \; -->}+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^5={\mathrm{\alpha <!--\; \alpha \; -->}}^2+\mathrm{\alpha <!--\; \alpha \; -->} \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^6={\mathrm{\alpha <!--\; \alpha \; -->}}^3+{\mathrm{\alpha <!--\; \alpha \; -->}}^2 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^7={\mathrm{\alpha <!--\; \alpha \; -->}}^3+\mathrm{\alpha <!--\; \alpha \; -->}+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^8={\mathrm{\alpha <!--\; \alpha \; -->}}^2+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^9={\mathrm{\alpha <!--\; \alpha \; -->}}^3+\mathrm{\alpha <!--\; \alpha \; -->} \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^{10}={\mathrm{\alpha <!--\; \alpha \; -->}}^2+\mathrm{\alpha <!--\; \alpha \; -->}+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^{11}={\mathrm{\alpha <!--\; \alpha \; -->}}^3+{\mathrm{\alpha <!--\; \alpha \; -->}}^2+\mathrm{\alpha <!--\; \alpha \; -->} \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^{12}={\mathrm{\alpha <!--\; \alpha \; -->}}^3+{\mathrm{\alpha <!--\; \alpha \; -->}}^2+\mathrm{\alpha <!--\; \alpha \; -->}+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^{13}={\mathrm{\alpha <!--\; \alpha \; -->}}^3+{\mathrm{\alpha <!--\; \alpha \; -->}}^2+1 \\ {\mathrm{\alpha <!--\; \alpha \; -->}}^{14}={\mathrm{\alpha <!--\; \alpha \; -->}}^3+1 \end{gathered}$$
Given the above element representation of $GF(2^4)$, we return to the enigmatic claim that for g(x)=x8+x4+x2+x+1, there is the root g(α)=0 over GF(24). We have:
$g(x)=x^8+x^4+x^2+x+1$
So it seems that for g(x)=x8+x4+x2+x+1 polynomial, $g(\mathrm{\alpha <!--\; \alpha \; -->})\ne <!--\; \ne \; -->0$.
Can anyone show me how to demonstrate that α (as a GF(16) element) is a root of a polynomial that was computed over GF(2)?
If it would be helpful, I can scan the 3 pages of the book and you can see the author's own words for the above ideas. He throws this information at you very quickly, so it's fairly difficult for me (as a non-mathematician) to follow.

How to learn Mathematics for Machine Learning?
Is the following path (with the following books) the correct way to learn Mathematics for Machine Learning? 0. Pre-algebra (Pre-Algebra Essentials For Dummies by Mark Zegarelli, Krista Fanning)
College Algebra (Schaum’s Outline of College Algebra by Robert E. Moyer, Murray R. Spiegel)
Pre-calculus (Schaum’s Outline of Precalculus by Fred Safier)
Calculus (Schaums Outline of Calculus by Frank Ayres, Elliott Mendelson)
Differential Equations (Schaum’s Outline of Differential Equations by Richard Bronson, Gabriel B. Costa)
Discrete Mathematical Structures (Schaums Outline of Discrete Mathematics by Seymour Lipschutz, Marc Lipson)
Linear Algebra (Introduction to Linear Algebra by Gilbert Strang)
Statistics and Probability (Practical Statistics for Data Scientists 50+ Essential Concepts Using R and Python by Peter Bruce, Andrew Bruce, Peter Gedeck)
Here's the lengthier version of my query. I was never bad with studies, however, I got bad grades and just passed my college because of some personal issues. I have been working since then in jobs which didn't require much mathematics.
I have switched careers now and I am working as a Machine Learning Engineer now. I did a course in Machine Learning but skipped the mathematics part. I want to study mathematics now, not just because it is required at work, but because I was always fascinated by mathematics. The issue is, I put in 60 hours a week in office. I take out time to study after work.
So I want to know if these books would actually help me understand the mathematics behind Machine Learning, or these will not cover it? Thank you for reading through this, any suggestion/help is highly appreciated.How to learn Mathematics for Machine Learning?

I have a 4-dimensional data in Excel as follows.
Row 1: Set a with 2 elements, /'Refinery 1','Refinery 2'/
Row 2: Set b with 2 elements, /'Pipeline','Rail'/
Column 1: Set q with 2 elements, /'Warehouse 1','Warehouse 2'/
Column 2: Set r with 3 elements, /'Petroleum','Diesel', 'Jet Fuel'/.
My data looks like this in Excel.
I try to import the data from Excel to GAMS by the following code.
sets a first row entries
b second row entries
q first column entries
r second column entries
parameter data3(a,b,q,r);
$CALL GDXXRW.EXE i=Try1.xlsx o=Try1.gdx set=a rng=Sheet1!A3:A6 rdim=1 set=b rng=Sheet1!B3:B6 rdim=1 set=q rng=Sheet1!C1:H1 cdim=1 set=r rng=Sheet1!C2:H2 cdim=1 par=data3 rng=Sheet1!A1:F6 Rdim=2 Cdim=2
$GDXIN Try1.gdx $LOAD a,b,q,r,data3 $GDXIN
display data3;
It shall produce the resulting parameter, i.e data3 as 24 elements (2x2x2x3), But it only shows 16 elements, due to some problems and ignores other elements.
The resulting GDX looks like this!
Any help will be highly appreciated. Thanks!

Sorry if this questions is overly simplistic. It's just something I haven't been able to figure out.
I've been reading through quite a few linear algebra books and have gone through the various methods of solving linear systems of equations, in particular, n systems in n unknowns. While I understand the techniques used to solve these for the most part, I don't understand how these situations present themselves. I was wondering if anyone could provide a simple real-world example or two from data analysis, finance, economics, etc. in which the problem they were working on led to a system of n equations in n unknowns. I don't need the solution worked out. I just need to know the problem that resulted in the system.

I am reading the book "Elements of Information Theory" by Cover and Thomas and I am having trouble understanding conceptually the various ideas.
For example, I know that H(X) can be interpreted as the average encoding length. But what does $H(Y|X)$ intuitively mean?
And what is mutual information? I read things like "It is the reduction in the uncertainty of one random variable due to the knowledge of the other". This doesn't mean anything to me as it doesn't help me explain in words why $I(X;Y)=H(Y)-<!--\; -\; -->H(Y|X)$. Or explain the chain rule for mutual information.
I also encountered the Data processing inequality explained as something that can be used to show that no clever manipulation of the data can improve the inferences that can be made from the data. If $X\to <!--\; \to \; -->Y\to <!--\; \to \; -->Z$ then $I(X;Y)\ge <!--\; \ge \; -->I(X;Z)$. If I had to explain this result to someone in words and explain why it should be intuitively true I would have absolutely no idea what to say. Even explaining how "data processing" is related to a markov chains and mutual information would baffle me.
I can imagine explaining a result in algebraic topology to someone since there is usually an intuitive geometric picture that can be drawn. But with information theory if I had to explain a result to someone at comparable level to a picture I would not be able to.
When I do problems its just abstract symbolic manipulations and trial and error. I am looking for an explanation (not these blah gives information about blah explanations) of the various terms that will make the solutions to problems appear in a meaningful way.
Right now I feel like someone trying to do algebraic topology purely symbolically without thinking about geometric pictures.
Is there a book that will help my curse?

Turbid water is muddy or cloudy water. Sunlight is necessary for most life forms; thus turbid water is considered a threat to wetland ecosystems. Passive filtration systems are commonly used to reduce turbidity in wetlands. Suspended solids are measured in mg/l. Is there a relation between input and output turbidity for a passive filtration system and, if so, is it statistically significant? At a wetlands environment in Illinois, the inlet and outlet turbidity of a passive filtration system have been measured. A random sample of measurements are shown below. (Reference: EPA Wetland Case Studies.)
Reading 1 2 3 4 5 6 7 8 9 10 11 12
Inlet (mg/l) 59.1 25.7 70.5 71.0 37.6 43.5 13.1 24.2 16.7 49.1 67.6 31.7
Outlet (mg/l) 18.2 14.3 15.3 17.5 13.1 8.0 4.1 4.4 4.3 5.8 16.3 7.1
Use a 1% level of significance to test the claim that there is a monotone relationship (either way) between the ranks of the inlet readings and outlet readings.
(a) Rank-order the inlet readings using 1 as the largest data value. Also rank-order the outlet readings using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test.
Reading Inlet
Rank x Oulet
Rank y d = x - y d2
1
2
3
4
5
6
7
8
9
10
11
12Σd2 =

Calculating cumulant from data set
Given a data set $x=\{x(n)\mid <!--\; \mid \; -->n=1,\dots <!--\; \dots \; -->,m\}$ how do I calculate the $p^{th}$-order cumulant? In particular I need to calculate the 4th-order cumulant. I found that I can calculate the cumulant from moments, but I also don't know how to calculate moments. I found how to calculate the 4th central moment (kurtosis) but I am not sure if/how they are related.
Since I have to implement the calculation of the cumulant in C++ I would be interested in a general explanation (to understand what I am doing) but also hints on how to implement the calculation efficiently would be appreciated. How can one interpret the 4th-order cumulant?
I am sorry if this is a trivial question but I don't have a strong mathematical background and was not able to find the answer online.

Interpreting a Computer Display. In Exercises 9–12, refer to the display obtained by using the paired data consisting of Florida registered boats (tens of thousands) and numbers of manatee deaths from encounters with boats in Florida for different recent years (from Data Set 10 in Appendix B). Along with the paired boat/manatee sample data, Stat-Crunch was also given the value of 85 (tens of thousands) boats to be used for predicting manatee fatalities.
StatCrunch Manatees = -49.048987 + 1.4062442 Boats
Sample size: 24
R (correlation ooefficient) = 0.85014394
Estimate of error standard deviation: 9.6605284
Predicted values:
X value Pred Y s.e.(Pred. y) 95% C.I. for mean 95% P.I. for new
85 70.481772 85 1.9724935(66.391071, 74.572473)(50.033706, 90.929839)

When we solve an equation, do we suppose that it is true and then work backward?
A couple of days ago I was reading Calculus by James Stewart and I read this:
Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you might be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x−5=7, we suppose that x is a number that satisfies 3x−5=7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x=4. Since each of these steps can be reversed, we have solved the problem.
This sounded strange to me! I have always thought that when we solve an equation we don't suppose that the equation is already satisfied. I have always thought that when we solve an equation we use algbraic property of numbers to obtain a simpler equation that is equivalent to the starting equation. And since the equations are equivalent we don't need to suppose that the initial equation is true, because when the last one is true, is true also the first one.
In other words: if I have to solve 3x−5=7 I don't need to suppose that x is a number that satisys 3x−5=7, I simply ad 5 to bothe sides to obtain the equivalent equation 3x=12, then I divide both sides by 3 to obtain the equivalent equation x=4, when the last one is true, is true also the first one and vice versa, the last one is true when x is replaced by 4, so 4 is the solution.
And so this is my question: is it true that when we solve an equation we implicitly suppose that x satisfies the equation (and we need to do that) to apply algebraic properties that give us the equivalent and simplier equation? Do we need this logic assumption in solving equations?
Thanks.
EDIT
Let me try to explain in a better way why I don't understand the need to suppose that x satisfies the equation (i.e. that x makes true the equality). Excuse me for the lenght of this edit.
Let's say I want to solve in R the equation 3x−5=7.
I know that
∀a,b,c∈R,a+c=b+c↔a=b(P1)
and I know that
∀a,b,c∈R,(c≠0→(ac=bc↔a=b))(P2)
Property P1 says to me that:
if a+c=b+c is true, then a=b is true;
if a=b is true, then a+c=b+c is true;
if a+c=b+c is false, then a=b is false;
if a=b is false, then a+c=b+c is false;
and property P2 says to me that, if c≠0, then
if ac=bc is true, then a=b ia true;
if a=b is true, then ac=bc is true;
if ac=bc is false, then a=b is false;
if a=b is false, then ac=bc is false;
If I know all of this, then starting from 3x−5=7 I don't need to suppose that x makes true the equality, because:
I can say that 3x−5=7 is equivalent to 3x=12 because of P1 without supposing that 3x−5=7 is true, they have the same truth value for the same value of x;
from 3x=12, I can say that it is equivalent to x=4 because of P2 without supposing that 3x=12 is true, they have the same truth value for the same value of x;
now I can say that 3x−5=12 is equivalent to x=4 without supposing that 3x−5=12 is true, they have the same truth value for the same value of x;
in the end I have the solution, because when x=4 is true, also 3x−5=7, so 4 is the solution.
What am I doing wrong? Why do I need to suppose that x satisfy the equation?When we solve an equation, do we suppose that it is true and then work backward?

I have been reading a paper: Wang X, Golbandi N, Bendersky M, Metzler D, Najork M. Position bias estimation for unbiased learning to rank in personal search.
I am wondering how they derive the M-step in an EM algorithm that has been applied to the problem in their paper.
This is the data log-likelihood function:
$\log <!--\; \; -->P(\mathcal{L})=\underset{(c,q,d,k)\in <!--\; \in \; -->\mathcal{L}}{\sum <!--\; \sum \; -->}c\log <!--\; \; -->{\mathrm{\theta <!--\; \theta \; -->}}_k{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}+(1-<!--\; -\; -->c)\log <!--\; \; -->(1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}).$
where observed data are (c:click, q:query, d:document, k:result position) rows, and ${\mathrm{\theta <!--\; \theta \; -->}}_k$ is the probability a search result at rank k is examined by the user (assumed to be independent of query-document pair), and ${\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}$ is the probability a search result, i.e., a query-document pair is actually relevant (assumed to be independent of result position).
Here are some hidden variable probability expressions (E means examination and R means relevant, they assume that a click event implies a search result is both examined and relevant):
$\begin{array}{l}P(E=1,R=1\mid <!--\; \mid \; -->C=1,q,d,k)=1\\ P(E=1,R=0\mid <!--\; \mid \; -->C=0,q,d,k)=\frac{{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)}(1-<!--\; -\; -->{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)})}{1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)}{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)}}\\ P(E=0,R=1\mid <!--\; \mid \; -->C=0,q,d,k)=\frac{(1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)}){\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)}}{1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)}{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)}}\\ P(E=0,R=0\mid <!--\; \mid \; -->C=0,q,d,k)=\frac{(1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)})(1-<!--\; -\; -->{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)})}{1-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t)}{\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t)}}\end{array}$
The paper has derived the M-step update formulas:
$\begin{array}{l}{\mathrm{\theta <!--\; \theta \; -->}}_k^{(t+1)}=\frac{\underset{c,q,d,k^{\mathrm{\prime <!--\; \prime \; -->}}}{\sum <!--\; \sum \; -->}{\mathbb{I}}_{k^{\mathrm{\prime <!--\; \prime \; -->}}=k}\cdot <!--\; \cdot \; -->(c+(1-<!--\; -\; -->c)P(E=1\mid <!--\; \mid \; -->c,q,d,k))}{\underset{c,q,d,k^{\mathrm{\prime <!--\; \prime \; -->}}}{\sum <!--\; \sum \; -->}{\mathbb{I}}_{k^{\mathrm{\prime <!--\; \prime \; -->}}=k}}\\ {\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}^{(t+1)}=\frac{\underset{c,q^{\mathrm{\prime <!--\; \prime \; -->}},d^{\mathrm{\prime <!--\; \prime \; -->}},k}{\sum <!--\; \sum \; -->}{\mathbb{I}}_{q^{\mathrm{\prime <!--\; \prime \; -->}}=q,d^{\mathrm{\prime <!--\; \prime \; -->}}=d}\cdot <!--\; \cdot \; -->(c+(1-<!--\; -\; -->c)P(R=1\mid <!--\; \mid \; -->c,q,d,k))}{\underset{c,q^{\mathrm{\prime <!--\; \prime \; -->}},d^{\mathrm{\prime <!--\; \prime \; -->}},k,}{\sum <!--\; \sum \; -->}{\mathbb{I}}_{q^{\mathrm{\prime <!--\; \prime \; -->}}=q,d^{\mathrm{\prime <!--\; \prime \; -->}}=d}}\end{array}$
however, how to get these two formulas?
Note: M-step is usually derived from this formula:
${\displaystyle Q(\mathit{\theta <!--\; \theta \; -->}\mid <!--\; \mid \; -->{\mathit{\theta <!--\; \theta \; -->}}^{(t)})={\mathrm{E}}_{\mathbf{Z}\mid <!--\; \mid \; -->\mathbf{X},{\mathit{\theta <!--\; \theta \; -->}}^{(t)}}<!--\; \; -->[\log <!--\; \; -->L(\mathit{\theta <!--\; \theta \; -->};\mathbf{X},\mathbf{Z})]}$
${\displaystyle {\mathit{\theta <!--\; \theta \; -->}}^{(t+1)}=\underset{\mathit{\theta <!--\; \theta \; -->}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}Q(\mathit{\theta <!--\; \theta \; -->}\mid <!--\; \mid \; -->{\mathit{\theta <!--\; \theta \; -->}}^{(t)})}$
where θ are parameters (i.e., ${\mathrm{\theta <!--\; \theta \; -->}}_k$ and ${\mathrm{\gamma <!--\; \gamma \; -->}}_{q,d}$ in this case), X are data (c,q,d,k in this case), and Z are hidden variables (E and R?).

Science of collecting, analyzing and interpreting data using mathematical procedures is called?

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.
Dunkin’ Donuts Service Times The normal quantile plot represents service times during the dinner hours at Dunkin’ Donuts (from Data Set 25 “Fast Food” in Appendix B).

Represent the following sentence as an algebraic expression, where "a number" is the letter x. A number is subtracted from 2.

Question. Vonna scored 27 goals at soccer practice. Her ratio of goals to misses was 3:7 how many times did she miss

The least common multiple of 12 and 17 is 6. True or false

Roger sells fire trucks and dolls to a toy store. He sells each fire truck for $15 and each doll for $10. Roger wants to sell at least 25 toys and needs to earn at least $300 to cover his costs. If each fire truck takes 50 minutes to make and each doll takes 45 minutes to make, how many fire trucks and dolls should Roger make to minimize the time spent making the toys.

The difference of a number p and -9 is 12. What is the number?

Maria's Pizza sells one 16-inch cheese pizza or two 10-inch cheese pizzas for $9.99. Determine which size given more pizza.