inequality – Probability that a system of linear inequalities of random variables is satisfied

Let $X_i$ (i = 1…n) be a set of independent uniform random variables. Is there a tool/method/concept to find the probability that a system of linear inequalities of $X_i$ is satisfied ?

$$
left{
begin{array}{c}
a_1 < b_{11} X_1 + b_{12} X_2 + … + b_{1n} X_n < c_1 \
… \
a_j < b_{j1} X_1 + b_{j2} X_2 + … + b_{jn} X_n < c_j \
… \
a_m < b_{m1} X_1 + b_{m2} X_2 + … + b_{mn} X_n < c_m
end{array}
right.
$$

with $a_j, b_{ji}, c_j in R$

For a single inequality, one can find the probability using convolution.

For several inequalities and $n < 4$, one can find the probability using a geometric method.

But what if $n geqslant 4$ ?

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probability – Sigma-algebra generated by random variable X proof

$(Omega, mathcal{F}, P)$ is probability space. If $X: Omega rightarrow mathbb{R}$ is random variable, the collection $S=left{ X^{-1} (B) | Binmathcal{B} right}$, where $mathcal{B}$ is Borel set, is a sigma algebra.

Since $mathbb{R}inmathcal{B}$ and $(forall Binmathcal{B})X^{-1}(B)inmathcal{F}$, because X is a random variable, we know that $X^{-1}(mathbb{R})=Omegainmathcal{F},$ but how do I prove that $Omega$ belongs to $S$?

Thank you.

FIPS and Random number generation using /dev/urandom on macOS [closed]

I am using /dev/urandom as an RNG. Does /dev/urandom confirm FIPS standards for Random Number generation?
Does it pass Continuous Random Number Generator Test too?

2 numbers are chosen at random from set ${1,2,3..5n}$ (without replacement). Find the probability that $n_1^4-n_2^4$ is divisible by 5

All the numbers can be written in the form of $5k,5k+1,5k+2,5k+3,5k+4$

Also the given expression can be written as
$$(n_1-n_2)(n_1+n_2)(n_1^2 + n_2^2)$$

Case 1

Let $n_1-n_2$ be a multiple of 5

Which implies that both numbers have to be of the from $5k$, ie. $n$ possible numbers

Case 2

Let $n_1+n_2$ be a multiple of 5

So $(n_1,n_2)$ can be $(5k,5k), (5k+1, 5k+4), (5k+4, 5k+1), (5k+2,5k+3), (5k+3, 5k+2)$

With $n$ cases for each ordered pair

Case 3

Let $n_1^2 + n_2^2$ be a multiple of 5

So here cases can be 5k and 5k only (IMO)

The cases with $(5k,5k)$ are repeated 3 times, so count that as one case only

So
$$P=frac {binom n2 + 2times binom n2 +2times binom n2}{binom {5n}{2}}$$

But that doesn’t match with the given answer $frac{17n-5}{5(5n-1)}$

Where am I going wrong?

android – Como podria hacer un Get Random para sacar imagenes de una URL

Estoy haciendo una app y quiero que en uno de los Fragment tenga un ImageView y que la imagen sea sacada de una URL que conecta a mi VPS. Me gustaría que me digan como puedo sacar una URL random de la carpeta img de la URL https://imagerandom.com/img(Esa URL es de ejemplo) y proyectarla en el ImageView

pr.probability – An Inequality of Expected Value of Random Variables

I encountered the following problem in my research:

Suppose there are $N$ random variables that are independent and identically distributed (IID). The probability density function (PDF) of these random variables $f(x)$ is a unimodal function symmetrical about $0$ (i.e., $f(x)$ is non-decreasing within $(-∞,0)$, and for any $x$, $f(x) = f(-x)$ holds. for example, the distribution can be uniform distribution, normal distribution, Cauchy distribution with mean $0$, etc.).
For a given real number $x_0$, Sort these random variables as $X_1, X_2, …, X_N$ such that $$|X_1-x_0|leq |X_2-x_0| leq … leq |X_N-x_0|$$
For example, if $N = 3$, the $N$ random variables are randomly chosen as $-0.5, 1.5, 5$, and $x_0 = 1$, then $X_1 = 1.5, X_2 = -0.5, X_3 = 5$.
Let $Y_i = |frac{X_1+X_2+…+X_i}{i}-x_0|^r (i=1,…,N, r = 1 or 2)$, then for any $x_0$ and $f(x)$, does the inequality
$$EY_1 leq EY_2 leq… leq EY_N$$
always hold? Where $E$ denotes the expected value.

The inequality above is tested via the Monte Carlo method for cases where the distributions are uniform distribution, normal distribution, and Cauchy distribution. Details can be seen in https://math.stackexchange.com/questions/4039555/an-inequality-of-expected-value-of-random-variables since I cannot post figures here…

Moreover, is it possible to derive the PDF of $Y_i$?

Answers or ideas for either $r=1$ or $r=2$ would be so grateful!

probability theory – First hitting time of a symmetric random walk

Definitions:

Let $xi_n$ be a symmetric random walk, i.e.,
$$
xi_n=eta_1+eta_2+dots+eta_n,
$$

where ${eta_n}$ is a sequence of i.i.d. random variables such that
$$
P{eta_n=1}=P{eta_n=-1}=frac{1}{2}.
$$

Furthermore, we define the first hitting time to be $$tau=minleft{n:|xi_n|=Kright},$$
where $K$ is a positive integer.


I was reading a book on stochastic processes and here we want to show that $tau<infty$ a.s. The book proves this as follows

We want to show that $P{tau=infty}=0.$ To this end we shall estimate $P{tau>2Kn}.$ Notice that $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^nto0$$ as $ntoinfty.$ Thus, we have begin{align}
P{tau=infty}&=bigcap_{n=1}^infty P{tau>2Kn} \ &=lim_{ntoinfty} P{tau>2Kn}=0.
end{align}

After spending so many time, I could not figure out how to get the inequality $$P{tau>2Kn}le left(1-frac{1}{2^{2K}}right)^n$$ in the first line of the proof. Can someone help me understand why this inequality holds?

Many thanks in advance.

time – Dúvida com Random e Atividades Periódicas em GoLang

Existe alguma forma de eu fazer com que seja executado um fmt.Printf(choosenCity) a cada 6 horas e, que a cada execução, ele escolha um elemento diferente do Array?
Eu até consegui fazer com que ele executasse de forma periódica depois de algumas gambiarras, mas li que não é recomendado usar time.sleep() e ele sempre resulta no mesmo valor, a não ser que eu feche e execute o script novamente.

arrCities := (6)string{
    "Teste1",
    "Teste2",
    "Teste3",
    "Teste4",
    "Teste5",
    "Teste6",
}
rand.Seed(time.Now().UnixNano())
choosenCity := arrCities(rand.Intn(len(arrCities)))
go func() {
        for true {
            fmt.Printf(choosenCity)
            time.Sleep(5 * time.Second)
        }
    }()
    // wait for 10 seconds before app finished
    time.Sleep(60 * time.Second)

}

Eu gostaria de (sem fechar o script), a cada 6 horas, pickar um elemento aleatório da Array, e então imprimir.

//OBS a Função `go func()` está dentro da main()

Any way to do a draw from a filtered SharePoint list (multiple random items)?

I have a list in SharePoint online where I want to filter on several columns and then select a specific number of random items that meet the filter criteria. Is this possible?

For example, I may have 3000 items in my list, but I want to choose some from the ones who have Status = Approved and School = Central High School. After I filter on the Status and School columns, I may have 75 list items. From those items, I need to define the number to be selected (let’s say in this case it was 50), and I’d like SharePoint to do a random selection of 50 items (from the 75 who met the filter criteria, out of the 3000 in the list). Ideally, I’d then like to use Flow to change the Status value for those list items, to change them from Approved to Placed.

I have a list where I’ll need to make multiple draws, where there may be 7-10 fields used to filter the list first.

My first thought was to do something manually (filter the list items, then have a “DrawNumber” column where I could assign each item a sequential number, then use a standalone RNG to pull from those numbers before manually reconciling the SharePoint list items to the RNG selections), but it will be time-consuming, so if I can automate it I’d prefer that.

If needed, I could create a view for each permutation and build some kind of flow around that, but what I’d love to do is be able to dynamically filter my results in the browser, choose the number of items to be picked, apply my “picker”, and then apply a flow to those items.

Another alternative that wouldn’t be too bad would be to generate the set of numbers in a standalone RNG if I could somehow paste them in (or otherwise access them) for the DrawNumber column filter.

So I could go to https://numbergenerator.org/ and set it to pull 50 numbers between 1 and 75, and then open that set of numbers in Excel:

Set of random numbers

But then, in my SharePoint list, the only way I see to choose the filter values is to click on each individual box (this screenshot shows ID, but I’d want to use my DrawNumber column):

SharePoint filter pane

Even if I create a view and try to paste in the numbers for the filter value, it only seems to accept a single number at a time (not a set of comma or space-delimited numbers). If I enter “1 2”, I get an error telling me that’s not a number. If I enter “1,2”, like this…

filter criteria

… it removes the comma and treats it like “12”.

Any suggestions on how to streamline this process?