2 votes
Accepted

Is the cosine angle between two R.V. an (approximation) not equality to the correlation coefficient?

Hint: $X \cdot Y$ is a random variable. $\text{Cov}(X,Y)$ is an expected value. ---- addendum ----* There is some confusion of terminolgy in your post. If $\bf X , \bf Y$ are two random vectors (in ...
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  • 33.5k
1 vote
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Compute expected value from two dimensional normal distribution

The term in the exponent of $f(x,y)$ can be written as $$\begin{split}-\frac 18\left[4x^2+8x(y+3)+6(y+3)^2\right] &= -\frac 12 \left[x^2+2x(y+3)+\frac 32(y+3)^2\right]\\ &=-\frac 12\begin{...
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  • 10.1k
1 vote
Accepted

When independent variables covary (not correlate) in the regression, why this happen?

It is because R cannot handle decimals that small. The model should be $y=1/999999iv1+1/999999iv2$ for model 2. If you do iv1=99999v1 and iv2=99999v2, you will get a significant interaction much fewer ...
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  • 10.1k
1 vote

Variance and covariance problem

I'm also getting that $\mathbb{Var}(X)\geq 2$. First of all, $$\begin{eqnarray*}2&=&\mathbb{Cov}(X,Y) \\&=& \mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y) \\&=& \mathbb{E}(XY)-\left(...
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  • 7,542
1 vote
Accepted

Covariance, and the Taylor expansion for the expected value of a linear function of random variables

Expectation is linear, and therefore $$ \mathbb{E}[\theta] = \mathbb{E}[U(X)]-\mathbb{E}[U(Y)] $$ regardless of whether $U,Y$ are independent or not. They could be arbitrarily correlated, this would ...
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  • 64.2k

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