Recent questions in Statistics and Probability

Z-Scores
Answered

Marcelo Maxwell
2022-09-27

Z-Scores
Answered

sombereki51
2022-09-27

a) True. That's what z scores are.

b) Not necessarily. Depends on the standard deviation in each of their classes.

Counting Principles
Answered

tamnicufl
2022-09-27

I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.

And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $\varphi $ function to prove it?

And can we also use some counting principle here to give an exact answer?

Study design
Answered

zakownikbj
2022-09-26

I came through this problem while studying for an exam in control systems:

Consider the following discrete time system

$$\overrightarrow{x}(k+1)=A\overrightarrow{x}(k)+b\overrightarrow{u}(k),\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{y}(k)=c\overrightarrow{x}(k)$$

where $b=(0,1{)}^{T},\phantom{\rule{thickmathspace}{0ex}}c=(1,0),\phantom{\rule{thickmathspace}{0ex}}A=\left[\begin{array}{ccc}2& & 1\\ 0& & -g\end{array}\right]$ for some $g\in \mathbb{R}$

Find a feedback regulation (if there is any) of the form $u(k)=-K\hat{x}(k)$ where $\hat{x}(k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e(k)=\overrightarrow{x}(k)-\hat{x}(k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.

My method

it is clean that the eigenvalues of the machine are ${\lambda}_{1}=2,{\lambda}_{2}=-g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $|z|=1$.

The state observer equation is

$$[\overrightarrow{x}(k+1)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k+1){]}^{T}=\left[\begin{array}{ccc}A-bK& & Bk\\ O& & A-LC\end{array}\right][\overrightarrow{x}(k)\phantom{\rule{thickmathspace}{0ex}}\overrightarrow{e}(k){]}^{T}$$

With characteristic equation

$$\chi (z)=|zI-A+bK|\phantom{\rule{thickmathspace}{0ex}}|zI-A+LC|={\chi}_{K}(z){\chi}_{L}(z)$$

Also consider

$$K=\left[\begin{array}{ccc}{k}_{1}& & {k}_{2}\\ {k}_{3}& & {k}_{4}\end{array}\right]$$

and let $a={k}_{1}+{k}_{3},\phantom{\rule{thickmathspace}{0ex}}\beta ={k}_{2}+{k}_{4}$

Then ${\chi}_{K}(z)=(z-2)(z+g+\beta )+a$.

So we can select some eigenvalues inside the unit circle and determine $a,\beta $ in terms of g. Choosing e.g. ${\lambda}_{1,2}=\pm 1/2$ we get $a=3g+33/8,\phantom{\rule{thickmathspace}{0ex}}\beta =9/4-g,\phantom{\rule{thickmathspace}{0ex}}g\in \mathbb{R}$

Questions

I want to ask the following:

Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $|{\lambda}_{1,2}|<1$?

Should I determine L matrix as well since the error must also vanish? (because it is not asked)

Correlation And Causation
Answered

Melina Barber
2022-09-26

Let X=(1,2,3,...,20). Suppose that $Y=({y}_{1},{y}_{2},...,{y}_{20})$ with ${y}_{i}={x}_{i}^{2}$ and $Z=({z}_{1},{z}_{2},...,{z}_{20})$ with ${z}_{i}={e}^{{x}_{i}}$. Pearson correlation coefficient is defined by formula

$\rho (X,Y)=\frac{\sum _{i=1}^{20}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sqrt{(\sum _{i=1}^{20}({x}_{i}-\overline{x}{)}^{2})(\sum _{i=1}^{20}({y}_{i}-\overline{y}{)}^{2})}}$

If $\rho (X,Y)=1$, we can say that X and Y have a linear correlation. If $0.7\le \rho (X,Y)<1$ then X and Y has a strong linear correlation, if $0.5\le \rho (X,Y)<0.7$ then X and Y has a modest linear correlation, and if $0\le \rho (X,Y)<0.5$ then X and Y has a weak linear correlation. Using this formula, we get $\rho $(X,Y)=0.9 and $\rho $(X,Z)=0.5. However, the relationship between X and Y is actually quadratic but they have the high correlation coefficient that indicate linear correlation.

So, my question is what is "linear correlation" actually between X and Y ? Since $\rho $(X,Z)=0.5 indicate the modest correlation coefficient, what is another intepretation of this value? What is the difference between $\rho $(X,Y) and $\rho $(X,Z), noting that Y and Z is not a linear function of X.

Sample Standard Deviation
Answered

Alan Sherman
2022-09-26

Percentile Rank
Answered

Elias Heath
2022-09-26

1st )

percentile ={ ( number of values below x + 0.5)/total number of observation } *100

2nd )

percentile ={ ( number of values below x )/total number of observation } *100

I have to know when to used 1st formula and when to used 2nd formula

propert explanation needed

Binomial probability
Answered

gemauert79
2022-09-26

The coefficient of x in the expansion of $(2x+4{)}^{5}$

640

5120

2560

1280

Normal distributions
Answered

hommequidort0h
2022-09-26

Correlation And Causation
Answered

malaana5k
2022-09-26

Can you explain to me what 'obligatory disclaimer' means in probability. Is it like a lack of information or what?

The context is the following:

Second, we found that the marginal distribution of Y is Bern(0.08), whereas the conditional distribution of Y given X = 1 is Bern(0.2) and the conditional distribution of Y given X = 0 is Bern(0.04). Since conditioning on the value of X alters the distribution of Y , X and Y are not independent: learning whether or not the sampled individual is a current smoker gives us information about the probability that he will develop lung cancer. This example comes with an obligatory disclaimer. Although we have found that X and Y are dependent, we cannot make conclusions about whether smoking causes lung cancer based on this association alone. (Joseph K. Blitzstein, Jessica Hwang--Introduction to Probability)

Mutually Exclusive Events
Answered

Conrad Beltran
2022-09-26

Compute the following:

$\sum _{i=1}^{k}P({A}_{i}^{c})$

Meta-Analysis
Answered

Medenovgj
2022-09-26

a. I don't know

b. Standard deviation

c. Effect sizes

d. Statistical significance

Bar Graphs
Answered

Jean Farrell
2022-09-25

Given the constraints (or non-constraints rather), is there a closed solution on a set of labeled vertices?

One way I looked at this problem is by trying to implement a constrained stars-and-bars technique on the diagonal and diagonal-exclusive half of the $n\times n$ adjacency matrix, but I'm not getting any good leads.

It seems there are multiple definitions of a self-loop in undirected graphs. In the context of this question, I define a self-loop to represent a degree of 2 in an undirected graph.

Percentile Rank
Answered

easternerjx
2022-09-25

Effect Size
Answered

Krish Crosby
2022-09-25

(a)small effect size

(b)medium effect size

(c)large effect size

Study design
Answered

basaltico00
2022-09-25

I'm teaching linear algebra to myself and in my study material I came upon notation that I just don't understand. I can't find an explanation for it from my material and it is hard to find on the internet as well it seems.

Example:

$U=L((3,2,-6,4),(0,4,1,-5))$

What does the L() notation mean? U itself should be a subspace for ${R}^{4}$. I would assume that those are vectors within the L().

Range
Answered

Zachariah Norris
2022-09-25

Correlation And Causation
Answered

Jazmyn Pugh
2022-09-25

Suppose I have a vector say:

[5 5 5 5 4 5]

then common sense says that there is a very high auto-correlation for the vector because it is more or less the same values. But when I try to calculate the auto-correlation coefficient, I'm getting a very low value(<0.3) for all lags. What does this mean? shouldn't it be higher because the series is very similar?Am I missing something?does correlatiom mean not similarity but similarity in rate(Rate of change)?

Correlation
Answered

deiluefniwf
2022-09-24

$\frac{\sum ({x}_{i}-\overline{x})({Y}_{i}-\overline{Y})}{\sqrt{\sum ({x}_{i}-\overline{x}{)}^{2}\sum ({Y}_{i}-\overline{Y}{)}^{2}}}$

If the only given values I have are $\sum {x}_{i},\sum {x}_{i}^{2},\sum {y}_{i},\sum {x}_{i}{y}_{i}$ is it even possible to compute the correlation coefficient?

Y-Intercept
Answered

Kelton Molina
2022-09-24

(a) $\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$

(b) $\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{\pi}{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$

(c) $\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\pi \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$

(d) $\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$

(e) $\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$

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