All Questions
Tagged with expected-value covariance
73
questions
1
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0
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57
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Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
- 345
7
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2
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799
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Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose
I'm taking a course in advance statistics and we have to prove whether the following expression is true: $E[zz^T]=E[z]E[z^T]$. I am assuming it is not, since the formula of the covariante matrix is $...
- 101
8
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2
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182
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Is there anything interesting to be taken from the fact that $E[(X-E[X])(Y-E[X])] = E[(X-E[X])(Y-E[Y])]$?
While playing around with the formula for covariance, I discovered something I wasn't expecting. Replacing the $E[Y]$ in the definition of covariance with an $E[X]$ appears to simplify back down to ...
- 223
1
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1
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94
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van der Vaart Asymptotic Statistics, page 38, why does $e_\theta'=\operatorname{Cov}_{\theta}t(X)$?
On Page 38 of van der Vaart's Asymptotic Statistics (near the bottom of the page), it says
By differentiating $E_\theta t(X)$ under the expectation sign (which is justified by the lemma), we see that ...
- 2,966
1
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1
answer
49
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Variance of $X + \alpha^\top Y$ where $X$ is a scalar random variable and $Y$ is a random vector [duplicate]
Consider a scalar random variable $X\in\mathbb{R}$, a vector random variable $Y\in\mathbb{R}^n$ and a constant (non-random) vector $\alpha\in\mathbb{R}^n$. I want to compute
$$
\mathbb{V}[X + \alpha^\...
- 657
1
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1
answer
38
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Prove covariance between sufficient statistic and logarithm of base measure in exponential family is equal to zero
Exponential family form is
$$f_X(x) = h(x)\exp(\eta(\theta)\cdot T(x) - A(\theta))$$
I know
$$\operatorname{Cov}(T(x), \log(h(x)) = 0.$$
But how can I prove it?
- 11
0
votes
1
answer
32
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Joint Hypothesis Testing-Variance
Got the following question:
Here is the provided answer:
I am confused about where the $9$ coefficient is coming from above. Any thoughts?
3
votes
1
answer
127
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Sign of Correlation between $X$ and $f(X)$ for strictly monotonic $f$
This question is a follow up to this question.
Suppose $f$ is strictly increasing. Can we say
$$\text{Cov}(X,f(X))\geq 0?$$
Ben's answer on the aforementioned linked post can be extended to show the ...
- 543
4
votes
1
answer
143
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Sign of Correlation between $X$ and $\log X$
Suppose $\text{supp}(X)\subseteq \mathbb{R}_{\geq 1}.$ Can we say $$\text{Cov}(X,\log X)\geq 0?$$
On one hand, we can say by monotonicity of log and Jensen's inequality that $$X\geq E[X]\implies \log ...
- 543
1
vote
1
answer
161
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Calculate expectation of a function with two dependent random variables
Hi Cross Validated community,
My question has to do regarding expectation of a multiplication of two random variables that are dependent.
Assume there are two random variables, one discrete: $G \in \{...
- 13
2
votes
1
answer
62
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Expectation given pairwise covariances
I have 4 variables A,B,C,D over {-1,1} (Rademacher variables) and know that ...
- 21
-1
votes
1
answer
53
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$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$
Take a random variable $X$. Is it true that
(1) $cov(X,f(X))\neq 0$ for any function $f$?
(2) $E(X f(X))\neq 0$ for any function $f$?
I believe the answer to both questions is no. However, can you ...
- 889
1
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0
answers
25
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mean and covarince matrix of AR(1) [closed]
assume I have a price data called pt, I fitted AR(1) model
p_t= alpha + beta pt_1 + e_t , ...
- 61
4
votes
1
answer
539
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some thought about independence and orthogonal, please comment on this if it's wrong
It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence.
In Statistics, the relation of ...
- 331
0
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0
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45
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Covariance of some random variables
I am given 2n-1 random variables, namely X₁, X₂... Xₙ, Xₙ₊₁... X₂ₙ₋₁. I also have E(Xᵢ)=𝜇 and Var(Xᵢ)=𝜎² for i=1,2,...2n-1. Suppose Y=X₁+X₂+...+Xₙ and W=Xₙ+Xₙ₊₁+...+X��ₙ₋₁ and I am asked to calculate ...
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