2 votes
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The gambler chooses dice at random, and rolls it $six$ times. What is the probability that fair die was chosen?

$$p(A|B) = \frac{p(A,B)}{p(B)}.$$ $A$ is the event that the die was fair. $B$ is the event that the $6$ die rolls turned out as specified. $$p(B) = p(A,B) + p(A^C,B).$$ $$p(A|B) = \frac{(1/2) \times (...
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2 votes

How to rigorously interpret and transform "equal chance" in different ways?

Putting the $100$ balls randomly in the $10$ boxes is equivalent to choosing randomly a mapping from the set of balls to the set of boxes. There are $10^{100}$ such mappings. The event «no box is ...
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2 votes
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Sequence of random variables converging to zero arbitrarily slowly

For any $\delta>0$ there exists $N_{\delta}\ge 1$ such that $$ \mathsf{P}(X_n\ge \delta)\le \delta $$ for all $n\ge N_{\delta}$. Thus, you may construct the required sequence $\{a_n\}$ as follows. ...
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1 vote

Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent.

Non-existence of non-constant independent random variables on this space: Let $X$ and $Y$ be independent random variables on this space and suppose they are both non-constant. Let $E$ be a non-empty ...
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1 vote

How to rigorously interpret and transform "equal chance" in different ways?

First of all, re Stars and Bars Theory, the two methods that you used do give the same answer. That is $p_{10}(100)$ bijects to determining the number of non-negative integer solutions to $x_1 + \...
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$(X,\Sigma,\mu)$ be a probability space, $A\subset\Sigma$ be a field s.t. $\Sigma=\sigma(A)$.Prove inf$\{\mu(E\Delta F):F\in A\}=0\forall E\in\Sigma$

Hints: Choose $F_n \in \mathcal A$ such that $\mu (E_n\Delta F_n)<\frac {\epsilon} {2^{n}}$. Check that $(\bigcup E_n) \Delta (\bigcup F_n )\subseteq \bigcup (E_n \Delta F_n)$. Of course, $\bigcup ...
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1 vote
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$(X,\Sigma,\mu)$ be a probability space, $A\subset\Sigma$ be a field s.t. $\Sigma=\sigma(A)$.Prove inf$\{\mu(E\Delta F):F\in A\}=0\forall E\in\Sigma$

I think one can prove directly that $\mathcal{M}$ is a $\sigma$-algebra. Given $\varepsilon>0$, If $A\in\mathcal{M}$ then there is $A'\in\mathcal{A}$ such that $\mu(A\triangle A')<\varepsilon$. ...
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1 vote
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Equivalence of a Levy measure

For $|x| \leq 1$, $$|x|^2 - \frac{|x|^2}{1 + |x|^2} = \frac{|x|^4}{1 + |x|^2} \leq \frac{|x|^2}{1 + |x|^2} \in L^1(\nu).$$ So the first integral is finite. For $|x| \geq 1$, $$1 - \frac{|x|^2}{1 + |x|^...
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1 vote
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Fubini's theorem and time integrals of stochastic processes

Your first application of Fubini's theorem is correct, but the second one does not seem to hold in general. First application Recall that to apply Fubini's theorem to $f$, you must have $f \in L^1( \...
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