Improve Your Understanding of College Level Statistics Problems

Which of the following describe discrete random variables, and which describe continuous random variables? a. The length of time that an exercise physiologist’s program takes to elevate her client’s heart rate to 140 beats per minute b. The number of crimes committed on a college campus per year c. The number of square feet of vacant office space in a large city d. The number of voters who favor a new tax proposal.

Sample size in Confidence Intervals
In repeating confidence interval experiments, are we allowed to take samples of different size every time? Because a confidence interval of 95% means that if the sampling process is repeated infinite times, 95% of all the intervals obtained will have the parameter of interest.
So wrt this process of repeating infinite times - do we have to repeat with the same sample size or can we vary it?

Show that, for random variables X and Z, $$E[(X-<!--\; -\; -->Y{)}^2]=E[X^2]-<!--\; -\; -->E[Y^2]$$ where Y=E[X|Z]

If X and Y are independent random variables with equal variances, find Cov(X+Y,X−Y).

Statistics and Confidence Intervals
Given the following set of values:
10,11,14,95,73,30,29,9,97,94,70
How do I calculate a 99% confidence interval for the sample mean? I am assuming that the variance is 10
Well, the idea I have is to assume that the distribution is normal, but after that i'm not completely sure what to do next. In particular, I am unable to find the z-score that corresponds to the z-score in the formula for the CI, i'm not sure what to input in R to find the CI. The formula for the confidence interval is:
$x-<!--\; -\; -->z_{\frac{a}{2}}\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}^2}{sqrt(n)}$ and $x+z_{\frac{a}{2}}\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}^2}{sqrt(n)}$ where x denotes the mean. Here the significance level (a) is 0.01.

I have two questions about the properties of binary set operations that I am having difficulty arriving at answers that I completely trust (though I am sure they are not difficult questions). Here they are:
1. Of the 16 binary operations on subsets satisfy the idempotent law? -Clearly the answer is at least 2, but binary operations P and Q are idempotent as well, correct?
2. Under how many of the 16 binary operations on the subsets of a set do the latter form a group? -I am not really sure how to solve this question.

Calculate the area of ​​a figure bounded by lines
x-y+3=0 x+y-1=0 y=0

I am trying to write a C++ program for parameter estimation(with Confidence Interval information) of an Exponentially distributed data set. I understand that $\mathrm{\lambda <!--\; \lambda \; -->}\stackrel{¯<!--\; ¯\; -->}{X}\sim <!--\; \sim \; -->\mathrm{\Gamma <!--\; \Gamma \; -->}(n,n)$. To come up with numeric values of lower/higher confidence intervals for a specified $\mathrm{\alpha <!--\; \alpha \; -->}$, I need to be able to compute Inverse Gamma. Could you please point me to some algorithms?

If X and Y are random variables, what is the meaning of E(XY) ?

Let X and Y be independent random variables distributed respectively as $${\mathrm{\chi <!--\; \chi \; -->}}^2(m)$$ and $${\mathrm{\chi <!--\; \chi \; -->}}^2(n)$$. Find the density of Z = X/(X + Y).

The joint density function for random variables X and Y is f(x,y) = {C(x+y) if $$0\le <!--\; \le \; -->x\le <!--\; \le \; -->3,0\le <!--\; \le \; -->y\le <!--\; \le \; -->2,0$$ otherwise. Find the value of the constant C.

Explain the difference between how probabilities may be represented for discrete and continuous random variables.

If the random variables x and y have joint density function, then?

Fuchsian Groups of the First Kind and Lattices
First, a Fuchsian group is a discrete subgroup of ${\text{PSL}}_2(\mathbb{R})$, which we can view as a group of transformations of the upper half-plane H that acts discontinuously. A lattice is a Fuchsian group with finite covolume. In other words, it is a Fuchsian group that has a fundamental domain in H with finite hyperbolic area (and then any two fundamental domains have the same hyperbolic area).
My confusion arises with the notion of a Fuchsian group of the first kind. Every definition I have seen for this term is roughly the same. A Fuchsian group Γ is said to be of the first kind if every point in $\mathbb{R}\cup <!--\; \cup \; -->\{\mathrm{\infty <!--\; \infty \; -->}\}$ (the boundary of H) is a limit point of the orbit Γz for some $z\in <!--\; \in \; -->\mathbb{H}$. Here, the notion of "limit point" is with respect to the topology on the Riemann sphere ${\mathbb{C}}_{\mathrm{\infty <!--\; \infty \; -->}}$.

Minimum number of weighings necessary
8 boxes each having different weights are numbered from 1 to 8 (the lightest 1, the heaviest 8). The total weight of 4 boxes are equal to the other 4’s total, and your task is to identify these two groups. You have a balance scale with two pans on which you can compare the weight of two groups each having exactly 4 boxes. What is the minimum number of weighings necessary to guarantee to accomplish this task?

How do you find the variance of the sum of two independent normally distributed random variables, X and Y, if the two variables are correlated? That is, Var(X+Y) = ___ ?

Pick balls of unequal weights from a given set of balls
There are 8 balls. Four of them weigh X grams each, and the other four weigh Y grams each. Your task is to find two balls having different weights. You have a balance scale with two pans on which you can compare the weight of any number of balls. What is the minimum number of weighings?

The joint probability mass function of the random variables X, Y, Z is p$$(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=\frac{1}{4}$$ Find E[XYZ].

This is a question from a sample mid-term for a first course in Statistics
Let p denote the proportion of elderly people with artificial hips. In an earlier study, the estimated proportion of the number of elderly people with artificial hips was found to be 11%.
Q.1. Using earlier study's estimate, find the number of samples we would need in order to create a 95 confidence interval for the proportion of elderly people with artificial which is no more than 0.05 away from the true proportion.
Ans. I have no clue how to do this. If could get the formula on how to approach this I would appreciate it.
Q.2. What is the maximum sample size needed to create a 93 confidence interval for the proportion of elderly people with artificial hips which is no more than 0.04 away from the true proportion ? Ans I would appreciate a formula for this as well.

Spectral sequence for Ext
If I have a morphism of schemes $f:X\to <!--\; \to \; -->Y$ and sheaves F,G on X, then is there a spectral sequence which relates the Ext-groups
Ext(f∗F,f∗G) on Y and Ext(F,G) on X?
I should add, that if this is not possible in general, that my morphism f is actually an affine morphism and that I want to compare the two Ext-groups in any way. The first thing I thought of was a spectral sequence, but perhaps there are other ways you know.