# 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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### Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
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### What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
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### hypothesis testing for 3$n$ samples where each $n$ sample has different underlying distribution

I have a question related to hypothesis testing as follows: I have $3n$ samples where the the first $n$ samples are drawn iid according to distribution $P_1$, second $n$ samples are drawn iid ...
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### Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...
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### What is the chance of collision of latter group of card number among different people?

Let's assume we are a payment system issuing 16-digit cards. If we have X customers and issue Y cards, how to calculate the chance of at least single collision of last 4 digits within a single ...