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Confused about parameters of a discrete SIR infectious disease model
In a discrete SIR infectious disease model:
n = time in days (as would happen in the case of covid-19).
$S_n$ = number of susceptible in day n
$I_n$= number of infectives in day n
$R_n$ = number of recovered (or removed) in day n
And,
$S_n-<!--\; -\; -->\frac{\mathrm{\beta <!--\; \beta \; -->}}{N}{S}_n{I}_n$
$I_{n+1}=I_n+\frac{\mathrm{\beta <!--\; \beta \; -->}}{N}{S}_n{I}_n-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}{I}_n$
$R_{n+1}=R_n+\mathrm{\gamma <!--\; \gamma \; -->}{I}_n$
Where;
$\mathrm{\beta <!--\; \beta \; -->}$ = infection rate (the number per day that susceptible people become infected)
$\mathrm{\gamma <!--\; \gamma \; -->}$ = recovery rate (the probability that an infected individual recovers). And therefore, $\frac{1}{\mathrm{\gamma <!--\; \gamma \; -->}}$ is the average length of the infectious period of the disease.
After simulating the SIR model and fitting it to available data using Excel's Solver (LSSE), the best parameter values were: $\mathrm{\beta <!--\; \beta \; -->}$ = 3.993 and $\mathrm{\gamma <!--\; \gamma \; -->}$ = 3.517.
This implies that, if this data were to model Covid-19 cases in some city, the number of susceptible people who become infected per day is 3.993 and that the average length of the infectious period of the disease is $\frac{1}{3.517}\approx <!--\; \approx \; -->0.2843$ days. This does not make sense to me. I applied the same approach to covid-19 cases in some city and the results were similar. Am I interpreting these parameters correctly? Are my definitions of the parameters correct? - sorry two questions, but asking about the same thing.
Thank you in advance for your help.

I have not been able to understand following text from book
There are one billion people who might be evil-doers. 2. Everyone goes to a hotel one day in 100. 3. A hotel holds 100 people. Hence, there are 100,000 hotels – enough to hold the 1% of a billion people who visit a hotel on any given day. 4. We shall examine hotel records for 1000 days
How do they come to conclusion that there are 100,000 hotels? How are 100,000 hotels enough to hold 1% of a billion people.
Then in the same example they have mentioned following
Thus, the chance that they will visit the same hotel onone given day is 10−9. The chance that they will visit the same hotel on two
different given days is the square of this number, 10−18. Note that the hotels
can be different on the two days.
This probability calculation is also not clear to me. Any help will great. Thanks

Intuition of information theory
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. $I(X;Y)\ge <!--\; \ge \; -->I(X;Z)$. 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?

I am reading Iwasawa's paper 'On Γ-extensions of Algebraic Number Fields'. On Page 189, he discussed the following data:
Let G be a compact group, X a closed normal subgroup of G such that X is a p-primary compact abelian group (meaning that X is an abelian pro-p group) and that G/N=Γ. Here Γ is topologically isomorphic to the additive group of ${\mathbb{Z}}_p$.
Iwasawa claimed that it is easy to see that the group extension G/X splits (first line of Page 190), but I don't see a proof. My knowledge on topological groups is limited, so I would appreciate it very much if anyone can provide me with a proof or some references.

Descriptive statistical analysis
I have some data obtained from questionnaire. The question asks to perform a descriptive statistical analysis for the above data and hence interpret your results. What does it mean? Is it mean I have to find the mean median mode standard deviation variance?

Is the Bayesian Prior representing a hypothesis with no data, or with all data?
I have an understanding question about the Bayes' Theorem: in
$p(z|x)=\frac{p(x|z)p(z)}{p(x)},$
the term p(z) is usually interpreted as the prior probability distribution of a hypothesis z before observing any data x.
However, if we write p(z) as the marginal
$p(z)=\int <!--\; \int \; -->p(z,x)dx=\int <!--\; \int \; -->p(z|x)p(x)dx={\mathbb{E}}_{x\sim <!--\; \sim \; -->p(x)}p(z|x),$
then the term p(z) seems to contain the knowledge about all data x.
Therefore, is the prior really representing the hypothesis with no data, or with all data?
We are not any smarter with all data than we are with no data?
Or is it a question of perspective?
How should I understand the prior correctly?
Thank you!

Clarifying Bayes' Rule
Looking through the literature dedicated to Bayes factors I noticed that given formulae are quite unclear from the point of Kolmogorov probability axioms.
E.g. consider the following expression from Wikipedia:
$Pr(M\mid <!--\; \mid \; -->D)={\displaystyle \frac{Pr(D\mid <!--\; \mid \; -->M)Pr(M)}{Pr(D)}},$
where D stands for our sampled data, M - for the model.
Well, D is a random variable. But how should I interpret M? In applications, e.g. solving a hierarchial probabilistic model I can consider Pr(D∣M) as the conditional density under given parameter. But the concept of Pr(M) seems rather misleading to me.
Are there any papers with formal definitions based on axiomatic approach?

Meaningful statistic measure of data pairs
I have a dilemma.
I have pairwise data, (a,b), that represents some form of speed, whether it's miles/hour or megabits/second. Let's say that we have the following set of data from measuring the "speed" of the same "car" on two different "courses" A and B under different conditions (say weather), ignoring the unit for now, and with apologies for my notations.
{(a,b)} = {(1, 4), (1, 2), (1, 1), (2, 1), (4, 1)}
There are a few ways to interpret the data:
1) If one believes that the "car" runs faster on "course" A and calculates the average of speed-up (a/b):
average speedup = (1/4 + 1/2 + 1/1 + 2/1 + 4/1)/5 = 1.55
Voila, "course" A makes the "car" go 55% faster than "course" B!
2) If one believes that the "car" runs faster on "course" B and calculates the average of speed-up (b/a):
average speedup = (4/1 + 2/1 + 1/1 + 1/2 + 1/4)/5 = 1.55
Voila, "course" B makes the "car" go 55% faster than "course" A!
3) If one believes say CDF is the right way to compare the data set, then it will be something like:
A: 1, 1, 1, 2, 4
B: 1, 1, 1, 2, 4
Voila, "courses" A and B are more or less identical.
Obviously, all three interpretations mask out important details, but at the same time, I am not able to come up with a good statistical measure to describe this set of data. This is clearly just a toy example, but I am facing similar dilemmas interpreting real data.
Thanks!

Examples of morphisms of schemes to keep in mind?
What are interesting and important examples of morphisms of schemes (especially varieties) to keep in mind when trying to understand a new concept or looking for a counterexamples?Examples of what I'm looking for:The projection from the hyperbola to the affine line has finite fibers but it is not a finite morphismThe Froebnius map is a morphism of varieties which is a bijection nevertheless not an isomorphismThe map (x,y)↦(xy,y) shows that the image of an affine variety need not be affineThanks for the excellent answers! To be a little more specific, I am especially interested in reasonable examples (so no line with infinitely many origins or the product of infinitely many fields). If the examples already make sense in the classical settings then all the better.

Interpreting confidence interval of regression coefficient.
In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure.
Here 95% confidence interval of regression coefficient, ${\mathrm{\beta <!--\; \beta \; -->}}_1$ is (.4268,.5914).
So i have interpreted as :
"The data provides much evidence to conclude that the true slope of the regression line lies between .4268 and .5914 at $\mathrm{\alpha <!--\; \alpha \; -->}=5$% level of significance."
But it is not understandable to those who don't know statistics.
How can i interpret it more generally?

Sample Variance in Principal Components Analysis
I was reading this Why is the eigenvector of a covariance matrix equal to a principal component?. And in the top answer, the poster mentions that if the covariance matrix of the original data points $x_i$ was $\sum <!--\; \sum \; -->$, the variance of the new data points is just $u^T\Sigma u$. I'm not sure why this is the case. Could someone enlighten me?

Bookkeeping is mainly concerned with
a. All options are correct.
b. Interpreting the data.
c. Recording the economic activities.
d. Designing the systems for recording, classifying and summarizing.

Boy Born on a Tuesday - is it just a language trick?
The following probability question appeared in an earlier thread:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The claim was that it is not actually a mathematical problem and it is only a language problem.
If one wanted to restate this problem formally the obvious way would be like so:
Definition: Sex is defined as an element of the set boy,girl.
Definition: Birthday is defined as an element of the set Monday,Tuesday,Wednesday,Thursday,Friday,Saturday,Sunday
Definition: A Child is defined to be an ordered pair: (sex $\times <!--\; \times \; -->$birthday).
Let (x,y) be a pair of children,
Define an auxiliary predicate $$\begin{gathered} H(s,b): \\ ! \\ !⟺<!--\; ⟺\; -->s=\text{boy}\text{ and }b=\text{Tuesday} \end{gathered}$$.
Calculate $P(x\text{ is a boy and }y\text{ is a boy}|H(x)\text{ or }H(y))$I don't see any other sensible way to formalize this question.
To actually solve this problem now requires no thought (infact it is thinking which leads us to guess incorrect answers), we just compute
$\begin{array}{rl}& P(x\text{ is a boy and }y\text{ is a boy}|H(x)\text{ or }H(y))\\ \\ =& \frac{P(x\text{ is a boy and }y\text{ is a boy and }(H(x)\text{ or }H(y)))}{P(H(x)\text{ or }H(y))}\\ \\ =& \frac{P((x\text{ is a boy and }y\text{ is a boy and }H(x))\text{ or }(x\text{ is a boy and }y\text{ is a boy and }H(y)))}{P(H(x))+P(H(y))-<!--\; -\; -->P(H(x))P(H(y))}\\ \\ =& \frac{\begin{array}{rl}& P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday})\\ \\ +& P(x\text{ is a boy and }y\text{ is a boy and }y\text{ born on Tuesday})\\ \\ -<!--\; -\; -->& P(x\text{ is a boy and }y\text{ is a boy and }x\text{ born on Tuesday and }y\text{ born on Tuesday})\\ \end{array}}{P(H(x))+P(H(y))-<!--\; -\; -->P(H(x))P(H(y))}\\ \\ =& \frac{1/2\cdot <!--\; \cdot \; -->1/2\cdot <!--\; \cdot \; -->1/7+1/2\cdot <!--\; \cdot \; -->1/2\cdot <!--\; \cdot \; -->1/7-<!--\; -\; -->1/2\cdot <!--\; \cdot \; -->1/2\cdot <!--\; \cdot \; -->1/7\cdot <!--\; \cdot \; -->1/7}{1/2\cdot <!--\; \cdot \; -->1/7+1/2\cdot <!--\; \cdot \; -->1/7-<!--\; -\; -->1/2\cdot <!--\; \cdot \; -->1/7\cdot <!--\; \cdot \; -->1/2\cdot <!--\; \cdot \; -->1/7}\\ \\ =& 13/27\end{array}$
Now what I am wondering is, does this refute the claim that this puzzle is just a language problem or add to it? Was there a lot of room for misinterpreting the questions which I just missed?

Solving a two-sample z-test for two population proportions is to be performed using the P-value approach.
The null hypothesis is $H_0:P_1=P_2$ and the alternative is $H_a:P_1\ne <!--\; \ne \; -->P_2$. Use the given sample data to find the P-value for the hypothesis test. Give an interpretation of the p-value.
$n_1=200$
$n_2=100$
$x_1=11$
$x_2=8$
I got a p-value of 0.4009 but I'm not sure how to interpret this p-value and what it means or if my result is correct.

How does big-O notation relate to the actual error involved in a numerical differentiation?
Suppose I have some position data $x_1,x_2,...x_n$ that was sampled at an interval h. If I wanted the velocity data, I could apply a finite difference scheme:
$v_1=\frac{x_2-<!--\; -\; -->x_1}{h}+O(h)$
O(h) denotes that the error term is proportional to the step size. ..What exactly does this mean physically? For instance, say my x terms were taken at 5 samples/second (so h=0.2). Does this mean the error in my velocity data is $\pm <!--\; \pm \; -->$ 0.2? How do I interpret the error in a physical sense, and can I write this like an uncertainty to a measurement? E.g.,
$v_1=10\pm <!--\; \pm \; -->0.2$?

Inconsistency in two-sided hypothesis testing
Suppose you have two sets of data with known population variances and want to test the null hypothesis that two means are equal, ie. $H_0:{\mathrm{\mu <!--\; \mu \; -->}}_1={\mathrm{\mu <!--\; \mu \; -->}}_2$ against $H_1:{\mathrm{\mu <!--\; \mu \; -->}}_1>{\mathrm{\mu <!--\; \mu \; -->}}_2$. There's a certain way I want to think about it, which is the following:
$\begin{array}{rl}P({\mathrm{\mu <!--\; \mu \; -->}}_1>{\mathrm{\mu <!--\; \mu \; -->}}_2)& =P(-<!--\; -\; -->({\mathrm{\mu <!--\; \mu \; -->}}_1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_2)<0)\\ & =P(\frac{{\stackrel{¯<!--\; ¯\; -->}{x}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{x}}_2-<!--\; -\; -->({\mathrm{\mu <!--\; \mu \; -->}}_1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_2)}{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\delta <!--\; \delta \; -->}\stackrel{¯<!--\; ¯\; -->}{x}}}<\frac{{\stackrel{¯<!--\; ¯\; -->}{x}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{x}}_2}{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\delta <!--\; \delta \; -->}\stackrel{¯<!--\; ¯\; -->}{x}}})\\ & =P(z<\frac{{\stackrel{¯<!--\; ¯\; -->}{x}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{x}}_2}{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\delta <!--\; \delta \; -->}\stackrel{¯<!--\; ¯\; -->}{x}}})\end{array}$
To me, this 'derivation' makes it perfectly clear what's actually going on. You're actually calculating the probability that H1 is true and not just blindly looking up some z-score.
However, now suppose that $H_1:{\mathrm{\mu <!--\; \mu \; -->}}_1\ne <!--\; \ne \; -->{\mathrm{\mu <!--\; \mu \; -->}}_2$. The problem with this is that the method I just described doesn't seem to work. If I write
$P({\mathrm{\mu <!--\; \mu \; -->}}_1\ne <!--\; \ne \; -->{\mathrm{\mu <!--\; \mu \; -->}}_2)=P({\mathrm{\mu <!--\; \mu \; -->}}_1<{\mathrm{\mu <!--\; \mu \; -->}}_2)+P({\mathrm{\mu <!--\; \mu \; -->}}_1>{\mathrm{\mu <!--\; \mu \; -->}}_2)$
Then all that happens is $P({\mathrm{\mu <!--\; \mu \; -->}}_1\ne <!--\; \ne \; -->{\mathrm{\mu <!--\; \mu \; -->}}_2)=1$. I think I'm probably not interpreting the above equation correctly.

Meaning of $\frac{P(X\cap <!--\; \cap \; -->Y)}{P(X)P(Y)}$
Imagine that we have a set $\mathrm{\Omega <!--\; \Omega \; -->}$ and X and Y are events that can happen, I mean, $P(X),P(Y)>0$. Then, what does it mean the ratio $\frac{P(X\cap <!--\; \cap \; -->Y)}{P(X)P(Y)}$?
I know that $\frac{P(X\cap <!--\; \cap \; -->Y)}{P(X)P(Y)}=\frac{P(X|Y)}{P(X)}=\frac{P(Y|X)}{P(Y)}$ and if that ratio is equal to 1 then X,Y are independent events, but I can't figure out what exactly it means... please give simple examples.
I found this when reading about lift-data mining.

Comparing Sample Mean and a Random Variable
Let $X_{(i)}=(i=1,2,\dots <!--\; \dots \; -->,n+1)$ be a random sample of size n+1 that is produced from a normal population. Let M be the sample mean of the first n random variables in this random sample.
Find the probability that
$P(X_{(n+1)}-<!--\; -\; -->M<0.5)$
Variance =1 and Sample Size n=40.
I have been stuck in this question for hours since I cannot find a formula to compute $X_{(n+1)}$. I tried computing the sample mean including (n+1) and then figure out something between two sample means but I went nowhere.
Am I interpreting this question wrong, isn't $X_{(n+1)}$ just a data value or does that notation mean its the sample mean of n+1 values? I cannot see how this problem could be solved if $X_{(n+1)}$ is a value ....

I have finished elementary probability and I know the sum of all probabilites in a data set is 1.But while reading Binomial Distribution,I encountered the formula for the Probability mass distribution :
$f(k;n,p)=\mathrm{Pr}<!--\; \; -->(K=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-<!--\; -\; -->p{)}^{n-<!--\; -\; -->k}$
Well, I know that probability is a fraction less than 1, and fractions multiplied with fractions will still yield a lesser fraction as the product. But what I am confused about is that the "$(\genfrac{}{}{0ex}{}{n}{k})$" that we multiply at the start of the formula is a positive integer, and I am confused that why shouldn't the net product be more than one?
I mean, the fraction that we get after multiplying the probabilities, wouldn't that turn greater than 1 if we multiply (which is repeated addition by itself) that fraction by the positive integer we get as a result of "$(\genfrac{}{}{0ex}{}{n}{k})$" ?
Sorry about my roundabout way of talking. You are only requested to prove that the whole thing is a value less than 1. I tried and just couldn't see why it shouldn't be greater than 1. I am in learner's stage (I could have learnt the forumla by rote, but my video instructor says if I proceed like that without understanding things and asking questions, then I will be like a monkey on a type-writer.

Can Pearson's Correlation coefficient distinguish between y=x and $y=\sqrt{x}$?
For a set of points (x,y), I obtained a Pearson's r of 0.9936004531.
For the same set of points, I changed them to $(\sqrt{x},y)$ (took the square root of every x value), and I obtained a Pearson's r of 0.9997411537, which is greater by 0.0061407006.
These data were obtained from an Physics experiment I did, where theoretically, $y\propto <!--\; \propto \; -->\sqrt{x}$.
How should I interpret this result?
I am leaning towards the idea that because the difference in Pearson's r is so small, we cannot conclude whether $y\propto <!--\; \propto \; -->\sqrt{x}$ or not, rather we can only vaguely conclude that as x increases, y increases in some fashion.
Thank you very much for the help..
Edit: There were only five data points, precise to 3 decimal places.
The five points:
(5,0.321)(10,0.395)(15,0.457)(20,0.510)(25,0.550)