# Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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### property of universally measurable function

Let $\mathcal B_d$ be the Borel $\sigma$-algebra on $\mathbb R^d$ and define $\mathcal B_d^*= \bigcap_{\rho} \overline{\mathcal B_d}^{\rho}$, where $\overline{\mathcal B_d}^{\rho}$ is the completion ...
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### Is there a way to write conditional expectation as an integral?

Let $E[X|G]$ be a random variable that is G-measurable and satisfies the partial averaging property, then we know that $E[X|G]$ is a conditional expectation. This is the definition I saw from Shreve's ...
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### Positive moments of independent variables are also independent

Suppose we have $X$ and $Y$ which are random variables and they are also independent and we also have $i, j \in \mathbb{N}_{+}$. Is it true that $X^{i}$ and $Y^{j}$ are independent? Actually I need ...
Is there a collection of random variables $X_1,X_2,\ldots,X_n$ such that $Y_1=X_1-X_2,~Y_2=X_2-X_3,\ldots,~Y_n=X_n-X_1$, are independently uniformly distributed on $[-1,1]$. How $X$'s should be ...
Consider two random variables $X$ and $Y$ defined on the same probability space $(\Omega,\sigma,P)$. We know that they are equivalent in the sense that $P(\{X \ne Y\})=0$. Let $A_X$ and $A_Y$ be the ...