**0 covariance questions.**

I have a stochastic process (Ornstein-Uhlenbeck) defined as:
$X(t) = e^{-at}(\int_0^t e^{a \tau} dW(\tau) + X_0)$
Where $W(t)$ is the Wiener process, and $X_0$ is the initial value of my process.
I ...

I want to make a model in which the dependent variable is nutritional levels of children, for this, I calculated $z$ scores like weight for ...

I am working with 30 data elements $x_i$ from a model. I consider the data $x_i$ to have zero error. What I want to do, is to fit a model to the following transformation of this data:
$y_i = y(x_{i-1}...

I am trying to estimate a function $f(x)$ at $x=0.1, 0.2, 0.3, 0.4, 0.5$.
I make this estimate through some complex procedure based on a set of independent but noisy data (with known uncertainties). ...

According to this page on wikipedia, if $X\sim N(\mu,\Sigma)$ with $\mu\in\mathbb R^2$ and $\Sigma\in\mathbb R^{2\times2}$ then we have $\textrm{Cov}(X_1,X_2)=0\implies X_1\perp\!\!\!\perp X_2$. Is ...

While working on an application of Gaussian process regression to ultrasound imaging, I came across an interesting similarity between the K-distribution and the Matérn covariance function.
Background
...

I know from previous studies that
$Var(A+B) = Var(A) + Var(B) + 2 Cov (A,B)$
However, I don't understand why that is. I can see that the effect will be to 'push up' the variance when A and B covary ...

I'm doing a nonlinear least-squares regression to find best-fit values for two parameters. I then want to use these best-fit parameters and their variances to extrapolate to a predicted value.
It's ...

I have a set of response processes (queue lengths in infinite server network).
Using queue theory, I can numerically calculate response autocovariance structure, from the known service time ...

Working with the generalized covariance formula for a vector $x$, I have the following:
$$E[(x-\mu)(x-\mu)^T)] = E(xx^T) - \mu E(x^T)$$
But the term $E(x^T)$ doesn't make much sense to me. Does ...

I am having trouble figuring this out. Any help would be appreciated.

In my setup,
there are $m$ trials.
Each trial has a probability $q$ of being selected.
$N \leq m $ is the number of selected trials
$$ \rightarrow N \sim \text{Bin}(q, m) $$
For each of the $N$ ...

I have been trying to find an expression for the covariance of two normally distributed variables X and Y if cov(x,y)=c then cov(x,xy)=?
I would greatly appreciate any help. Probably it must be ...

We are trying to quantify synchrony in water chemistry variation among several thousand sites. For each site we have a time-series of concentration. We'd like to quantify the overall temporal ...

We have an ARMA process as below:
$$
X[n] = a_0 Z[n] + a_1 Z[n-1] + b_1 X[n-1] +b_2 X[n-2].
$$
How can I find the mean and covariance of the process to show that it is/is not a weakly stationary ...

I have the returns of three stocks, $R_{1t}$, $R_{2t}$, $R_{3t}$, with 100 monthly observations for each return series. Lets suppose that I create a portfolio consisting of stocks 1 and 2, $P_t=w_{1t}...

I have model
$$Y = \beta + \epsilon$$
where $Y$ is scalar and $\beta \in \mathbb{R}$. $\epsilon$ has mean $0$ and variance $\sigma^2$.
If I perform a k-fold cross validation, what is the ...

I am not able to understand logic behind coming up with this formula for covariance. We know that the sample covariance formula is:
$${\rm Cov}(x,y)=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}$$
...

I'm trying to calculate a covariance matrix using weighted data in a single pass, and I'm not sure that I'm doing it correctly. I found a wikipedia article which gave the following python* code:<...

I have a dataset of vectors represented by a $n_{rows} \times n_{cols}$ matrix $M$. Each row is a vector and each column is a feature. There are $n_{cols} = n \times n$ features.
We use the sample ...

I am not sure what the formula is for the covariance of an AR(2) process
$$
X_t = \phi_1 X_{t−1} + \phi_2 X_{t−2} + \epsilon_t
$$
where $\{\epsilon_t\}$ is white noise process (Gaussian) $N(0, \sigma^...

Say $X \in \mathbb{R}^n$ is a random variable with covariance $\Sigma \in \mathbb{R}^{n\times n}$. By definition, entries of the covariance matrix are covariances:
$$
\Sigma_{ij} = Cov( X_i,X_j).
$$
...

Questions
Does it depend on whether the tree is shallow or deep? Or can we say this irrespective of the depth/levels of the tree?
Why is bias low & variance high? Please explain intuitively and ...

Suppose $y$ is a continuous random variable and $d$ is a binary random variable that takes the value $1$ with probability $p$ and $0$ with probability $1-p$.
How do I show that $\text{Cov}(y,d)=(E[y|...

Say we have a random variable $Y$ which is positive, and a binary variable $X$, i.e. takes values 0 and 1 only.
I have seen (in various places) that:
$$Cov(X,Y) = E[Y | X = 1| - E[Y | X = 0|$$
How ...

For a pair of generic random variables $(A, B)$, I am trying to find a way to express
$$E[A^2B]$$
using only the marginal moments
$$E[A^k], E[B^k], k=1,2,\ldots $$ and the second joint moment
$$...

Is there a reason why someone would estimate a VAR and then get the estimates of covariance of the data, instead of just using the usual sample covariance estimate formula for each observable variable?...

Say $X_t$ is a time series. Find the autocovariance function for $\Delta X_t$ using explicit methods.
I get:
$$ Cov(X_{t}-X_{t-1},X_{t-k}-X_{t-1-k}))=Cov(X_t,X_{t-k}) - Cov(X_t,X_{t-1-k})-Cov(X_{t-...

I am interested in the co-variance of two joint coefficients, with one variable appearing in either joint coefficient, i.e. $\mathrm{Cov}[x+z,x+w]$.
But let me start with my model, let's say I want ...

For random variables $X,Y\in\mathbb{R}$ we say that they are orthogonal if $E(XY)=0$ and uncorrelated if $E((X-E(X))(Y-E(Y))=0$. In what follows I assume all random variables to be centered so ...

In simple linear regression $y=\beta_0+\beta_1x$, you the least square estimator $\hat\beta_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}$ can also be represented as $\hat\beta_1=\frac{cov(...

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