Questions tagged [covariance]

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

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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: ...
2answers
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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. ...
4answers
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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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3answers
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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 ...
1answer
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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 ...
2answers
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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 $E(X^n) = \frac{n!}{\lambda^n}$...
1answer
271 views

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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3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
1answer
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Prove that Cov(X,Y)=Cov(X,E[Y|X])

I've been working on this problem for 3 hours now, and my complete lack of progress is getting disheartening. I've looked up definitions, proofs, and have even seen a solution for this particular ...
1answer
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A question about a generalization of covariance

Suppose, $H$ is a Hilbert space over $\mathbb{R}$. Suppose, $X$ and $Y$ are random vectors in $H$. Let’s define Hilbert expectation of a random vector $X$ in a Hilbert space $H$ as a vector $v \in H$, ...
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Degrees of Freedom in Covariance: Intuition?

If we say $\operatorname{Var}(x)$ has $n-1$ degrees of freedom which are lost after we estimate $\operatorname{Var}(x)$, this matches how $n-1$ observations are now constrained to be sufficiently ...