# Tagged Questions

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### Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
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### Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
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### Is monotonicity condition not required in this short derivation?

For given density functions $p_1(x)$ and $p_0(x)$ ($x\in\mathbb{R}$) the following equation is to be satisfied: $$(1-\epsilon_1)\{P_1[p_1/p_0>c] +cP_0[p_1/p_0\leq c]\}=1$$ where $c\in\mathbb{R}^+$ ...
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### For a given random variable X, does there exists another random variable Y in different probability space, X and Y have the same distribution

X (known) is a random variable defined on probability space $(Î©_1 ,F_1 ,P_1 )$ , there exists a random variable Y in another probability space $(Î©_2 ,F_2 ,P_2)$ which has the same distribution as X. ...
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### Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $(\Omega, \mathscr{A}, \mu )$ ...