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How to Calculate $\frac{1}{\sqrt{n}}x^{\top}y$ given random vectors $x$ and $y$?

Assuming $X$ have finite second moment, $$Z_n:=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_i Y_i$$ has zero mean and variance $\mathbf{Var}(Z_n)=\mathbf{E}[X^2]$. So, By the central limit theorem, $Z_n$ ...
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1 vote

MIT Statistic For Applications course Question 1

It's Markov's inequality. thanks Matthew Towers.
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Accepted

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Creating a martingale given

You are correct, the filtration by construction makes the process adapted. You also need to check the integrability condition of the definition. To check the martingale property, you need to use the ...
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1 vote

Sigma field generated by gaussian random variable

$\sigma(X)=\sigma(Y)$ iff $X$ is measurable function of $Y$ and conversely. So any independent gaussian variables $X$ and $Y$ are such that $\sigma(X)\neq\sigma(Y)$.
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Expectation of a Standard Normal Random Variable

I see the question is already answered, so I'll present a trick that can help later. I'll break the problem into two sections, even and odd powers, this allows us to use symmetry arguments. Odd Power ...

"Random" generation of rotation matrices

If you have MATLAB, I suggest using the code from What Is a Random Orthogonal Matrix? by the late Professor Nicholas Higham (n in code represents the dimension). <...
1 vote
Accepted

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1 vote
Accepted

Find a continuous increasing function $T:(0,1)\to (0,\infty)$ s.t. $Y=T(X)$ has p.d.f $g(y)=\frac{2}{(y+1)^3}$

You were nearly done: $$g(y)=\frac{d}{dy}[T^{-1}(y)]$$ $$\implies \frac{-1}{(y+1)^2}-\frac{-1}{(0+1)^2}=T^{-1}(y)$$ $$\implies \frac{-1}{(T(x)+1)^2}+1=x$$ $$\implies T(x)=\frac1{\sqrt{1-x}}-1.$$
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Tightness of random variables and Borel-Cantelli Lemma

It's a definition of the term "tight". There is no "fact" in the definition, so there is nothing to "prove". A sequence $(X_n)_{n=1}^{\infty}$ is said to be tight if the ...
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A standard Laplace variable has the form $X_1-Y_1$ where $X_1$ and $Y_1$ are iid with density $e^{-x}1_{(0,\infty)}(x)$. The sum of $n$ iid standard Laplace variables has the form $S_n=X_n-Y_n$ where ...