# Questions tagged [covariance]

Questions about covariance, a measure of (linear) association between two random variables.

1,098 questions
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### Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
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### What does Determinant of Covariance Matrix give?

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. ...
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### When does the inverse of a covariance matrix exist?

We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see Covariance matrix). We also know that every symmetric positive ...
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### Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
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### Is a symmetric positive definite matrix always diagonally dominant?

A Hermitian diagonally dominant matrix $A$ with real non-negative diagonal entries is positive semidefinite. Is it possible to have a Hermitian matrix be positive semidefinite/definite and not be ...
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### Prove that $U=Y - E[Y|X]$ and $X$ are uncorrelated

Let $U = Y - E[Y|X]$. How can I prove that $U$ and $X$ are not correlated? I've been doing a lot of things but when I calculate $\text{cov}(U,X)$ I finish with $EXY - EXEY$ and not $0$ which would be ...
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### Variance of Z = max(X,Y) where X Y are jointly bivariate normal

I have a question about the bivariate normal r.v.'s Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$. My ...
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### Covariance of $X^2$ and $X^3$ when $X$ is exponentially distributed

Here is my work.... \begin{align*} Cov(Y,Z) &= E(YZ) - E(Y)E(Z)\\ &= E(X^2\cdot X^3) - E(X^2)E(X^3)\\ &= E(X^5) - E(X^2)E(X^3) \end{align*} And we know $E(X^n) = \frac{n!}{\lambda^n}$...
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### Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
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### Compute $E(X\mid X+Y)$ if $(X,Y)$ is centered normal with known covariance matrix [closed]

The random variable $(X,Y)$ has a two dimensional normal distribution with mean $(0,0)$ and covariance matrix $\begin{pmatrix} 4&2 \\ 2&2 \end{pmatrix}$. Find $E(X\mid X+Y)$. I am ...
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### Bound the variance of the product of two random varables.

For two random variables $X$ and $Y$ show that the following inequality holds $$\mathrm{Var}(XY)\leq 2\|Y\|_{\infty}^{2}\mathrm{Var}(X)+2\|X\|_{\infty}^{2}\mathrm{Var}(Y).$$ Well first I tried to ...
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### What linear transformations preserve these conditions?

Main Question Let's define $\Gamma(n)$ as the set of real antisymmetric matrices of size $n$ ($n$ is an even Integer), fulfilling: $$\forall \gamma\in \Gamma(n) \Rightarrow \gamma^2=-\mathbb I_n$$ ...
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### For two random variables $X_1 + X_2$ and $\min(X_1,X_2)$ find the joint-distribution and the covariance

Let $X_1,X_2$ be independent random variables. Moreover $X_1,X_2$ are discrete uniform distributed({$1,...,N$}) We define: $A:= X_1+X_2$ $B:= \min(X_1,X_2)$ Find joint-distribution of $A$ and $B$...
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### Given a set of 3 orthogonal vectors, how can I find a minimum volume enclosing ellipsoid expressed in the Cartesian coordinate frame?

Generalized Problem Given values to start the problem: A 3D orthonormal coordinate frame (we'll call it the 'V' coordinate frame) that is rotated from the global coordinate system (we'll call it 'G') ...
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### Why does the variance formula has a square term?

I was reading about variance from Head First Statistics : And then - Q. I find the reasoning a little absurd. Wouldn't just taking the absolute distance suffice if cancelling out of the terms was ...