Questions tagged [covariance]

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

1,129 questions
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Is the positive part of a covariance stationary process also stationary?

I am wondering if it is possible to derive a result on the stationarity of the positive or negative part of a covariance stationary process. Namely, consider $\{ X_t \}, t=1,2,3,...,$ a covariance ...
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Does $\operatorname{Cov}(X,Y) = 0$ mean $\operatorname{Cov}(X,\log Y) = 0$?

Suppose $X,Y$ are positive random variables with $\operatorname{Cov}(X,Y)=0$. Define $Z= \log Y$. Does it necessarily follow that $\operatorname{Cov}(X,Z) = 0$? I know it's true for linear ...
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Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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Why does the identity $\mathbb{E}(X) = \mathbb{E}\left(\int \mathbb{1}_{u \leq X}du\right)$ hold?

I'm reading on Hoeffding's covariance identity, the proof of which is neatly covered here, or, in a similar manner, in this MSE post, but I can't seem to fully understand the trick/property used there....
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Variance of X - Y

If X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal. Using the information given, I was able ...
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Gaussian kernels for arbitrary metric spaces

Let $(I,d)$ be an arbitrary (pseudo-)metric space. Define the function $$c(i,i') := \exp\big( - d(i,i')^2 / 2 \big)$$ Is $c$ necessarily nonnegative-definite, hence a kernel function?
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mean and variance normalization of vectors

I have vectors $x \in \mathbb{R}^n$ and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that $y = M (x - b)$ has mean zero ($\operatorname{E}[Y] = 0$) ...
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Find covariance of a random process

Let $Y_t = e^{-\alpha t}W_{\beta \ \exp({2\alpha t})}$, where $W_{s} \ \text {is Wiener prosses,} \quad 0\le t, \quad \alpha , \beta\in \mathbb R^1$. Find $\text{Cov}(Y_t , Y_s).$ Here is my ...
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Show that the autocovariance function depends on $s$ and $t$ only through their difference $\left|s-t\right|$

Consider the time series $$x_t = \beta_1 + \beta_2t + w_t,$$ where $\beta_1, \beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2_w$. I want to show that the ...
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Finding the probability of two random variables being equal to 1

Question: A die is thrown $n+2$ times. After each throw a '$+$' is recorded for $4$, $5$, or $6$ and '$-$' for $1$,$2$, or $3$, the signs forming an ordered sequence. To each, except the first and ...
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finding the common variance of a die throw

would appreciate your help with this question: a regular die is being thrown 21 times. we define: $x_1$ - the number of throws we obtained 1 or 2. $x_2$ - the number of throws we obtained 3,4,5,6. ...
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calculate $\operatorname{cov}(X,Y)$ from $f_{x,y}(x,y)$

I have the following density function: $$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$ We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$...
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Central limit theorem for dependent random variables with covariance condition

Consider a sequence of identically distributed real-valued random variables $(X_i)_{i\in\mathbb{N}}$, with $\mathbb{E}\left[X_i\right]=0$ and $\mathbb{E}\left[X_i^2\right]=1$. Suppose that there ...
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Correlation function for ARMA($2$,$1$)

I have to derive analytic expressions for the autocorrelation function of this ARMA($2$,$1$) process: $$y_{t} = \varphi_{1}y_{t-1} + \varphi_{2}y_{t-2} + \varepsilon_{t} + \theta\varepsilon_{t-1}$$ ...
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Normal distribution, uncorrelated random variables

$Y$ is a random variable such that $$Y = \cases{X,\hspace{2mm} \mathrm{for}\hspace{1mm} |X| \leq c \\ \\-X,\hspace{2mm}\mathrm{for}\hspace{1mm}|X|>c,}$$ where $c \geq 0,$ and $X$ has the standard ...
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Covariance between centered and scaled normal entries of a random vector

From this post and following a tip by @Ian in the comments. If $X_1,\dots,X_n \sim \text{ i.i.d. } N(\mu,\sigma^2)$ with $\displaystyle \bar X= \frac{\sum_{i=1}^n X_i}{n},$ the covariance of the ...
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Variance of Chi Square Distribution as the Sum of Unit Normal Random Variables

Okay, so I am interested if there is a way to derive the variance for a Chi-Square distribution using the property that it is the sum of independent unit normal distributions squared. For example, ...
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Proof that sum over autocorrelations is -1/2

I am trying to understand a proof that shows, that the sum over the autocorrelation starting with lag=1 is always equal -1/2, for a stationay time series. The sum looks like this:  S_{\rm{afc}}=\...