## Expected value of a random variable conditioned on a positively correlated event

I have a random variable $$x \in [a, b]$$ with PDF $$f(x)$$ and an event $$E$$ which satisfies the following property for any $$x'.

$$\Pr[E|x > x’] \geq \Pr[E]$$

My question is whether or not the following inequality holds.

$$\int_{a}^{b} uf(u)\Pr[E|x=u]du \geq \Pr[E]\int_{a}^{b} uf(u)du$$

## Probability that maximal elements has the same position in samples from correlated random variables

Let $$x$$ and $$y$$ be two correlated random variable (say, standard normal) with correlation coefficient $$\rho>0$$. Let $$X= \{x_1, x_2, …, x_L\}$$ and $$Y= \{y_1, y_2, .. y_L\}$$ be samples of size $$L$$ from $$x$$ and $$y$$ respectively.

What is the probability that $$\mbox{argmax}\ X = \mbox{argmax}\ Y$$.

Alternatively, suppose that $$x_1$$ is maximal element, what is the probability that $$y_1$$ is maximal too.

Any references pointing to the solution of either question would also be appreciated.

## Multidimensional Correlated Geometric Brownian Motion, finding exact form of the matrices

My goal is to understand the dimensions of the matrices involved, so I am initially writing things as column vectors, and defining all the dimensions.

I am working with the following setup: Probability space $$(\Omega, \mathcal F, \mathbb Q)$$, equipped with a $$(d \times 1)$$-dimensional Correlated Brownian Motion, $$W$$, and the natural filtration of $$W$$ is $$(\mathcal F)_s$$.

The martingale, $$X$$, (with respect to $$\mathcal F_t$$ and $$\mathbb Q$$) is $$(d \times 1)$$-dimensional and of the form: $$\begin{equation} dX_t^i = \sigma_t^i X_t^idW_t^i, \: i \in [1,d], \qquad d\langle W^i, W^j \rangle_t = \rho^{i,j}_tdt \end{equation}$$

I have been trying to find the correct matrix form for this equation, but whenever I have looked online, the equation seems to always be written in the above form for each $$i$$, rather than as the matrices themselves.

So far, I have defined the $$(d \times d)$$ covariance matrix $$\Sigma$$, and another $$(d \times d)$$ matrix $$A$$: $$\begin{equation} AA^T \equiv \Sigma, \qquad \Sigma_{i,j} = \rho^{i,j}\sigma^i\sigma^j \end{equation}$$ and a $$(d \times 1)$$-dimensional standard Brownian Motion, $$B$$, and a $$(d \times 1)$$-dimensional vector $$L$$, so that : $$\begin{equation} \frac{dX_t^i}{X_t^i} \equiv L_i \end{equation}$$

So now, I have that: $$\begin{equation} L = AdB \end{equation}$$ I am not sure if this is correct, but it seems to contain all the relevant information. The covariances between each $$\frac{dX_t^i}{X_t^i}$$ is found through $$\Sigma$$ as $$\rho^{i,j}\sigma^i\sigma^j = \text{Cov}(\frac{dX_t^i}{X_t^i}, \frac{dX_t^j}{X_t^j})$$, so I think it should be correct.

From there I tried to convert $$L$$ to the $$(d \times 1)$$ dimensional vector $$dX$$, by multiplying by the diagonal $$(d \times d)$$ matrix $$D = \text{diag}(X_t^1,X_t^2,…)$$, which leads to:

$$\begin{equation} DL = dX = DAdB \end{equation}$$

I assumed this would work, and tried to check by using Ito’s Lemma on both $$dX_t^i = \sigma_t^i X_t^idW_t^i, \: i \in [1,d]$$, and on $$dX_t = DAdB_t$$, to check and the results seem to match.

I am using this form of Ito’s Lemma: \begin{align} df = \frac{\partial f}{\partial t}dt + \sum_i\frac{\partial f}{\partial x_i}dx_i + \frac{1}{2}\sum_{i,j}\frac{\partial^2 f}{\partial x_ix_j}[dx_i,dx_j] \end{align} I was just calculating the $$\frac{1}{2}\sum_{i,j}\frac{\partial^2 f}{\partial x_ix_j}[dx_i,dx_j]$$ term, so using $$dX_t^i = \sigma_t^i X_t^idW_t^i, \: i \in [1,d]$$ results in $$\frac{1}{2}\sum_{i,j}^d\frac{\partial^2 f}{\partial x_ix_j}\rho^{i,j}\sigma^i\sigma^jX^iX^jdt$$, as expected.

For the form $$dX_t = DAdB_t$$, I used that $$\frac{1}{2}\sum_{i,j}\frac{\partial^2 f}{\partial x_ix_j}[dx_i,dx_j] = \frac{1}{2}\sum_{i,j}(\beta\beta^T)_{i,j}\frac{\partial^2 f}{\partial x_i \partial x_j} dt$$, for any Ito process of the form $$dY_t = \beta dB_t$$.

This gives $$\begin{equation} \frac{1}{2}\sum_{i,j}(DA(DA)^T)_{i,j}\frac{\partial^2 f}{\partial x_i \partial x_j}dt = \frac{1}{2}\sum_{i,j}^d(D\Sigma D)_{i,j}\frac{\partial^2 f}{\partial x_i \partial x_j}dt = \quad \frac{1}{2}\sum_{i,j}^d(D_{i,i}\Sigma_{i,j} D_{j,j})\frac{\partial^2 f}{\partial x_i \partial x_j}dt = \frac{1}{2}\sum_{i,j}^d\frac{\partial^2 f}{\partial x_ix_j}\rho^{i,j}\sigma^i\sigma^jX^iX^jdt \end{equation}$$

I am wondering if this is correct, or if I did something incorrectly here. The dimensions seem to match everywhere. Is it possible to find a solution, like in this post: https://mathoverflow.net/questions/285251/solution-of-multivariate-geometric-brownian-motion. I can’t seem to get to that point using the form $$dX_t = DAdB_t$$.

Thanks a lot for the help!

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## Any algorithm to align two correlated random sequences the corresponding members of which are aligned

Two random sequences $$A = {a_1,a_2,a_3,…..a_i,….},B = {b_1,b_2,b_3,…..b_i,….}$$,are correlated, but the corresponding position or Subscripts of one are consistently move forward, or backward.for a special case ignoring randomness, $$A ={1,2,3,…..n,…}$$,and $$B ={2,4,6,…..2n,….}$$, is dislocated as $$B ={6,…..2j,….}$$.Is there an Any algorithm to align two correlated random sequences? And the computational complexity?

## Non-Negative irreducible matrices with random (correlated or independent) non-zero entries

Lets $$M$$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $$M$$, $$\lambda$$, is positive and equal to its spectral radius $$\rho(M)$$.

Now assume the matrix $$M$$ is not deterministic and its nonzero elements are equal to random variables $$\tanh(x_i)$$ with $$x_i\sim N(m>0, \sigma^2)$$. However, the zero elements are the same deterministic zeros as before. My question is that what will happen to the expected value of the maximum eigenvalue if $$x_i$$s are correlated compared to the case where they are independent.

My observation is that existence of positive correlation among the non-zero entries increases the expected maximum eigenvalue compared to the case where the entries are independent. But I am not able to justify this experiment.

## Is it possible to replace count distinct inside Correlated sub-queries?

I’m trying to optimize the query bellow. The execution Plan is showing multiple sorting before aggregations (I think this is due to the count distincts in the correlated sub-queries)…

Is it possible to minimize the number of sorting by doing it after the aggregations or at least replacing the count distinct by another function? Query:

 SELECT   "R"."Lat" AS "Lat", "R"."Long R" AS "Long R", "R"."Dept" AS "Dept", "R"."Reg" AS "Reg",  "B3"."M" AS "M", "B3"."St" AS "St", "B3"."Sp"AS "Sp", "B3"."Reg" AS "Reg", "B3"."year" AS "year", "B3"."id_program" AS "id_program", "B3"."program" AS "program", "B3"."Lib" AS "Lib", "B3"."Ef"/"R2"."NB Dept" AS "Agg_Ef", "B3"."Eex"/"R2"."NB Dept" AS "Agg_Eex", "B3"."Ein"/"R2"."NB Dept" AS "Agg_Ein", "B3"."Sehr"/"R2"."NB Dept" AS "Agg_Sehr", "B3"."Sin"/"R2"."NB Dept" AS "Agg_Sin", "B3"."Sr"/"R2"."NB Dept" AS "Agg_Sr", "B3"."Sc"/"R2"."NB Dept" AS "Agg_Sr",  "C2"."RE"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_RE", "C2"."RD"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_RD", "C2"."RDP"/("R2"."NB Dept"*"B2"."NB St")AS "Agg_RDP", "C2"."RC"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_RC", "C2"."RA"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_RA", "C2"."EE"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_EE", "C2"."BE"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_BE", "C2"."BD"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_BD", "C2"."BDP"/("R2"."NB Dept"*"B2"."NB St")AS "Agg_BDP", "C2"."BC"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_BC", "C2"."BA"/("R2"."NB Dept"*"B2"."NB St") AS "Agg_BA"  FROM "Database"."R_Table" "R"  INNER JOIN (  SELECT  "R"."Reg" AS "Reg",  COUNT (DISTINCT "R"."Dpt") AS "NB Dept" FROM "Database"."R_Table" "R"    GROUP BY "R"."Reg") "R2" on "R2"."Reg" = "R"."Reg"  INNER JOIN (  SELECT  "B"."Reg" AS "Reg",  COUNT (DISTINCT "B"."St") AS "NB St" FROM "Database"."B_Table" "B"    GROUP BY "B"."Reg") "B2" on "B2"."Reg" = "R"."Reg"  INNER JOIN (  SELECT "B"."Job" AS "Job", "B"."St" AS "St", "B"."Sp" AS "Sp", "B"."Reg" AS "Reg", "B"."year" AS "year", "B"."id_program" AS "id_program", "B"."program" AS "program", "B"."Lib" AS "Lib", SUM("B"."Ef") AS "Ef", SUM("B"."Eex") AS "Eex", SUM("B"."Ein") AS "Ein", SUM("B"."Sehr") AS "Sehr", SUM("B"."Sin") AS "Sin", SUM("B"."Sr") AS "Sr", SUM("B"."Sc") AS "Sc" FROM "Database"."B_Table" "B" GROUP BY "B"."id_program","B"."program","B"."year","B"."Reg","B"."Job",          "B"."Sp","B"."Lib","B"."St") "B3" on "B3"."Reg" = "R"."Reg"  LEFT JOIN ( SELECT "C"."Job" AS "Job", "C"."Sp" AS "Sp", "C"."Reg" AS "Reg", "C"."year" AS "year", "C"."id_program" AS "id_program", "C"."program" AS "program", "C"."Lib" AS "Lib", SUM("C"."RE") AS "RE", SUM("C"."RD") AS "RE", SUM("C"."RDP") AS "RDP", SUM("C"."RC") AS "RC", SUM("C"."RA") AS "RA", SUM("C"."EE") AS "EE", SUM("C"."BE") AS "BE", SUM("C"."BD") AS "BD", SUM("C"."BDP") AS "BPE", SUM("C"."BC") AS "BC", SUM("C"."BA") AS "BA" FROM "Database"."Content" "C" GROUP BY "C"."id_program","C"."program","C"."year",          "C"."Reg","C"."Job", "C"."Sp","C"."Lib") "C2" on            concat("C2"."id_program","C2"."program","C2"."year","C2"."Reg","C2"."Job","C2"."Sp","C2"."Lib") =           concat("B3"."id_program","B3"."program","B3"."year","B3"."Reg","B3"."Job","B3"."Sp","B3"."Lib") 

## Bound on the mutual information between a product of correlated random variables

Let $$G$$ be a finite group.

Suppose the random variables $$X_1,\dots,X_N$$ are sampled uniformly at random from $$G$$. Let $$Y_1,\dots,Y_N$$ be random variables where $$Y_i$$ is correlated with $$X_i$$ and sampled according to some unknown distribution.

Given a bound on the mutual information $$I(X_k:Y_k) \leq \epsilon_k$$ for all $$k$$, what is a good upper bound on $$I(X_1\dots X_N: Y_1\dots Y_N)$$? I.e., the product of the two sets of random variables.

I believe a bound like $$I(X_1\dots X_N: Y_1\dots Y_N) \leq C\prod\epsilon_k$$ for some $$C$$ might exist but have had no luck in proving it.

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## What analysis / algorithm helps stabilizing the fit of correlated parameters (but not colinear)?

I have many curves that I want to fit using a convolution of some functions. These functions include Weibull distributions with 2 parameters lambda and k, as well as a function B(t) such as measured curves to fit to model = F(lambda1, k1, kambda2, k2) + B(t)

The main problem here is that even if the lambda’s, k’s and B are not colinear, they can be “kind of” substituted and the optimization can lead to different local minima, with a close final error, but parameters not close at all.

This is a problem because I intend to interpret the value of these parameters as natural characteristics of the objects I observe.

Our actual approach is to minimize the number of parameters, i.e. fixing some of the lambda’s and k’s, as we would do if there were a function linking them. However this is arbitrary + this is a sacrifice as I can not interpret this parameters value anymore.

So question : is there a method / analysis / related problem / science paper dealing with this problem of unstable optimization when parameters are not exactly perpendicular degrees of liberty ?

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## A limit for two correlated variables

Suppose we have two correlated Normal variables $$X_A$$ and $$X_B$$, with respective standard deviation $$A$$ and $$B$$, and correlation $$\rho$$. The variable $$X_A + X_B$$ has a standard deviation in excess (or deficiency) of that of $$X_A$$ given by: $$E = \sqrt{A^2 + 2 \rho A B + B^2} – A$$ My question is: as $$A$$ becomes large (with $$A >> B$$), show rigorously that $$E \rightarrow \rho B$$.

This is straightforward to verify for $$\rho = \pm 1$$, but the intermediate case eludes me.

## Bootloader/BIOS, flashing ROM and correlated risks. Why Android devices are more brickable than PCs?

I have a solid experience of installing different OS (Linux, Windows,…) on PC. I would like to try just for fun to install Linux on a unbranded Android low cost tablet acquired in 2015. I spent some time browsing the web and as far as I understood there is a risk that during the flash procedure the device could be potentially damaged. So I read extensively on how to backup the ROM using TWRP and all related matters. I would like just to have some explanations on the below topic:

Scenario #1:
I have a PC, if I want to try another OS I can just format the hard disk and install it, in no way there is the risk of damaging the BIOS motherboard. Motherboard and hard-disk are separated, so no problems may arise.

Scenario #2:
I have a tablet, want to wipe out Android and install an upgraded version of Android or a Linux distro suitable for mobile devices.

• Why in this scenario there is a risk to get an unusable device?
• Is this because in this case motherboard and unit memory are bundled together? So wiping the memory will also erase the configuration settings of the motherboard?
• Do we have here the equivalent of BIOS settings?

Thanks to everybody who will like to explain

Ghera

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