Master Statistics and Probability Problems with Expert Help

Review of Bronfenbrenner's Ecological Systems Theory essay

1. A sample is chosen randomly from a population model that was strongly skewed to the right with m =145 and s =100 .
a. If we draw a random sample of n = 1000 from this population, what will be the shape of the histogram based this sample? What would be your best guess of mean and standard deviation of this sample?
b. If we draw 10,000 samples of size n = 1000 from this population, what will be the shape of the histogram of the sample means. What would be your best guess of sample mean and sample standard deviation of the distribution represented by this histogram?

Find the period of a 100ft building with special steel moment frames

How do I know if a Binomial model is appropriate?
I have a question which is about the number of weeks out of 5 in which an event occurs. I have a frequency table with a sample of 40 - with $x=0,1,2,3,4,5$ and freq, 2,7,11,12,6,2.
I have worked out the unbiased population mean and estimate - but then I'm not sure whether binomial what I need or not? I have to decide if a bionomial model is appropriate.
I can see that the data is discrete but its not binary like "event happens" or "event does not happen". It seems relatively symmetrical - and almost normally distributed? I'm not really sure how to work this out? Is a binomial model right or not?

When we take $X_1,...,X_n$ from $F(x)$. Then the ranks are $X_{(1)}<...<X_{(n)}$ and lets take the test-statistic,
$\begin{array}{r}{t}_0(X_1,...,X_n)=\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{X}_{(i)}\end{array}$
If we look at $t(-<!--\; -\; -->X_1,...,-<!--\; -\; -->X_n)$ to see if $t(-<!--\; -\; -->X_1,...,-<!--\; -\; -->X_n)\stackrel{?}{=}-<!--\; -\; -->t(X_1,...,X_n)$, does the rank change?
For example is then $-<!--\; -\; -->X_1,...,-<!--\; -\; -->X_n$ correspond to $X_{(1)}<...<X_{(n)}$?

A quick question regarding hypothesis testing (decision variable)
If we have two normally distributed populations $P_1$ and $P_2$, then to test the hypothesis $H_0:{\mathrm{\mu <!--\; \mu \; -->}}_1={\mathrm{\mu <!--\; \mu \; -->}}_2$ against an alternative hypothesis, we choose the decision variable according to three cases:
Case 1: If ${\mathrm{\sigma <!--\; \sigma \; -->}}_1$ and ${\mathrm{\sigma <!--\; \sigma \; -->}}_2$ are known, then regardless whether $n_1$ and $n_2$ are large or small, we consider the standard normal statistic:
$Z=\frac{{\stackrel{¯<!--\; ¯\; -->}{X}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{X}}_2}{\sqrt{\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_1^2}{n_1}+\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_2^2}{n_2}}}$
Case 2: If the $\mathrm{\sigma <!--\; \sigma \; -->}$'s are unknown, but $n_1$ and $n_2$ are large, then we consider:
$Z=\frac{{\stackrel{¯<!--\; ¯\; -->}{X}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{X}}_2}{\sqrt{\frac{S_{X_1}^2}{n_1}+\frac{S_{X_2}^2}{n_2}}}$
Case 3: If the $\mathrm{\sigma <!--\; \sigma \; -->}$'s are unknown and $n_1,n_2$ are small, then we consider the student-t statistic:
$T=\frac{{\stackrel{¯<!--\; ¯\; -->}{X}}_1-<!--\; -\; -->{\stackrel{¯<!--\; ¯\; -->}{X}}_2}{\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; \sigma \; -->}}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
Where ${\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; \sigma \; -->}}}^2:=\frac{(n_1-<!--\; -\; -->1)S_{X_1}^2+(n_2-<!--\; -\; -->1)S_{X_2}^2}{n_1+n_2-<!--\; -\; -->2}$
Is this correct?

How do you find the exact value of $\sec <!--\; \; -->({\tan }^{-<!--\; -\; -->1}<!--\; \; -->(-<!--\; -\; -->\frac{3}{5}))$?

Let $X_1,X_2,...,X_n$ be a random sample from a $N({\mathrm{\theta <!--\; \theta \; -->}}_1,{\mathrm{\theta <!--\; \theta \; -->}}_2)$ distribution. Find the uniformly minimum variance unbiased estimator of $3{{\mathrm{\theta <!--\; \theta \; -->}}_2}^2$.

Perform hypothesis test for population mean It is claimed that average rice production by Company2 is 209,500 cwt (Company2 = 209500.0). Test this claim using a hypothesis test at 1% level of significance.
My answer: Null = 209500.0 alternative = .01
I'm being told that these my answers are not correct. I have no idea why. Any help would be appreciated!

Consider a random sample of size n$n$ that follows a density probability function given by:
$f(x,\mathrm{\theta <!--\; \theta \; -->})=\frac{1}{\mathrm{\theta <!--\; \theta \; -->}}{x}^{-<!--\; -\; -->\frac{\mathrm{\theta <!--\; \theta \; -->}+1}{\mathrm{\theta <!--\; \theta \; -->}}}{\mathbb{1}}_{(1,+\mathrm{\infty <!--\; \infty \; -->})},\mathrm{\theta <!--\; \theta \; -->}>0$
where $\mathrm{\theta <!--\; \theta \; -->}$ is unknown.

Is the posterior always a compromise between the prior and the data?
Suppose that we are interested in learning the proportion of the population $\mathrm{\theta <!--\; \theta \; -->}$ with a particular property (for instance, the fraction of the population who are male). Suppose that we randomly sample n members of this population (with replacement, to make things easier) and observe that y of them have the property (so the fraction of the sample with the property is y/n). We start with a continuous prior p($\mathrm{\theta <!--\; \theta \; -->}$) with full support [0,1] and update this using Bayes rule.
Question: does the expected value of the posterior always lie between the prior expectation and the sample fraction y/n?

Given $X_1,..X_n$ observations of a population with densitiy
$f(x)=\{\begin{array}{c}\frac{1}{b-<!--\; -\; -->a}\text{ if }x\in <!--\; \in \; -->(a,b)\\ 0\text{ else }\end{array}$
where interval limits a and b are unknown. Determine the maximum likelihood estimations for a and b.
This is example task from exam and I like to know how solve it correctly.
When I understand formula of maximum likelihood estimation correct,
This is maximize when $(b-<!--\; -\; -->a)$ is as small as possible but also important that (a,b) include all the data. For this reason we have
$a=min(x_1,..,x_n)$
and
$b=max(x_1,..,x_n)$

On the generalized Sierpinski space
In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If $S=\{0,1\}$, then the Sierpiński topology on S is the collection $\{\varphi ,\{1\},\{0,1\}\}$ such that $\mathrm{\varphi <!--\; \varphi \; -->}\subset <!--\; \subset \; -->\{0\}\subset <!--\; \subset \; -->\{0,1\}$ we can generalize it by defining a topology analogous to Sierpiński topology with nested open sets on any arbitrary non-empty set as follows: Let X be a non-empty set and I a collection of some nested subsets of X indexed by a linearly ordered set $(\mathrm{\Lambda <!--\; \Lambda \; -->},\le <!--\; \le \; -->)$ such that I always contains the void set $\varphi$ and the whole set X, i.e.
$I=\{\mathrm{\varnothing <!--\; \varnothing \; -->},A_{\mathrm{\lambda <!--\; \lambda \; -->}},X:A_{\mathrm{\lambda <!--\; \lambda \; -->}}\subset <!--\; \subset \; -->X,\mathrm{\lambda <!--\; \lambda \; -->}\in <!--\; \in \; -->\mathrm{\Lambda <!--\; \Lambda \; -->}\}$
such that $A_{\mathrm{\mu <!--\; \mu \; -->}}\subset A_{\mathrm{\nu <!--\; \nu \; -->}}$ whenever $\mathrm{\mu <!--\; \mu \; -->}\le <!--\; \le \; -->\mathrm{\nu <!--\; \nu \; -->}$.
Then it is easy to show that I qualifies as a topology on X.
My questions are:
(1) Is there a name for such a topology in general topology literature?
(2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?

Hello, Plainmath. Help me to answer a question. Thanks
Explain why you are more likely to obtain a representative sample with proportionate stratified random sampling than with either stratified random sampling or simple random sampling.

Is the invariant subspace problem open for invertible maps?
Let $T:H\to <!--\; \to \; -->H$ be a bounded linear operator with bounded inverse on the separable complex Hilbert space. Does T preserve a closed proper non-trival invariant subspace?
I'm aware the question is (famously) open for bounded linear maps, and of partial results, but no survey (or Tao's blog, etc) seem to address the invertible case.
If it is open, does a positive or negative answer imply the answer in the non-invertible case?

Estimating standard deviation from wheighted sample
The standard deviation is given by $\sqrt{\frac{\sum <!--\; \sum \; -->(x_i-<!--\; -\; -->x{)}^2}{n}}$, however when we estimate the standart deviation from a sample, the best estimation is $\sqrt{\frac{\sum <!--\; \sum \; -->(x_i-<!--\; -\; -->x{)}^2}{n-<!--\; -\; -->1}}$
How do I have to adjust the standarddeviation if I want to wheight my samples?
I.e. the standard deviation would be $\sqrt{\frac{\sum <!--\; \sum \; -->w(x_i)(x_i-<!--\; -\; -->x{)}^2}{\sum <!--\; \sum \; -->w(x_i)}}$, if I had the entire data set. What is the correct estimation of the standard deviation, if I'm only given a subsample of the population?

Homotopy equivalence between two mapping tori of compositions
For any maps $s:X\to <!--\; \to \; -->K$ there is defined a homotopy equivalence
$T(d\circ <!--\; \circ \; -->s:X\to <!--\; \to \; -->X)\to <!--\; \to \; -->T(s\circ <!--\; \circ \; -->d:K\to <!--\; \to \; -->K);(x,t)\mapsto <!--\; \mapsto \; -->(s(x),t).$
Here, T(f) denotes the mapping torus of a self-map $f:Z\to <!--\; \to \; -->Z$ (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map:
$T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s),(k,t)\mapsto <!--\; \mapsto \; -->(d(k),t).$
If the above is a genuine homotopy inverse, then the map:
$(x,t)\mapsto <!--\; \mapsto \; -->(d(s(x)),t)$
would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are:
Is the map $T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s)$ I've defined above actually a homotopy inverse? If so, what is the homotopy from the composition I wrote down above to the identity map?
Is there a better one that makes the homotopy obvious?

Use the information at the beginning of the lesson. The Palermo scale value of any object can be found using the equation
$PS={\log }_{10}<!--\; \; -->R$
where R is the relative risk posed by the object. Write an equation in exponential form for the inverse of the function.

Generalized likelihood ratio statistic for two binomial distributions
This question develops hypothesis tests for the difference between two population proportions.Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: $H_0:p_1=p_2,H_A:p_1<p_{2}or{p}_1>p_2$
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y.