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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2 votes
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Expected value of $\bar x^2$ shows different values when calculating through different equations.

The line after "Also using the Estimator" is incorrect. It should be $$E(\bar x^2) = E\left(\left(\frac{1}{n}\sum_{i=1}^n x_i\right)^2\right) = E\left(\left(\frac{1}{n}\sum_{i=1}^n x_i\right)...
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  • 2,156
1 vote

How to find central moment from characteristic function

$$E[e^{izW_1}] = \cos(\sqrt{2iz})^{-1/2} = \cosh(\sqrt{2z})^{-1/2} = \Big( \sum_{n=0}^{+\infty} \frac{(2z)^n}{(2n)!}\Big)^{-1/2}.$$ This shows that the characteristic function is analytic and enables ...
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1 vote

Expected value and variance of $Z=-\ln(1-F(X))$

For any random variable $X$ which has an invertible distribution $F$, which is the case for continuous r.v., $U=F(X)$ is uniform on [0,1], always. Thus, the density of $U$ is $f_U(u) = 1_{(0,1)}(u)$ ...
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  • 145
1 vote
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Rough estimation for $L^2$ norm of sum of random variables

Fix any $\lambda >0$. Let $X_1,X_2$ be independent where $\operatorname{Var}[X_1]=1$ and $\operatorname{Var}[X_2]=\lambda$, and $X_3=-X_2$. We have $$ \mathbb{E}[(X_1+X_2+X_3)^2] = \|X_1\|_2^2 = 1 $...
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