# 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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### Forming a triangle probabilistically [duplicate]

What is the probability that if you break a stick at $2$ points the three sides form a triangle? Is there a technique that avoids calculus?
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### $X-x_0=O_p(n^{-1/2})$ implies $g(X)-g(x_0)=O_p(n^{-1/2})$

Suppose that $X$ is a random vector and $x_0$ is a fixed vector such that $$X-x_0=O_p(n^{-1/2}).\tag{*}$$ Let $Y=g(X)$ where $g$ has a continuous gradient that is nonzero at $x_0$. Let $y_0=g(x_0)$...
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### $X_i = \mathcal{N}(0, \sigma_i^2)$, with $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$

Let $\{X_k\}$ be independent random variables such that $X_i = \mathcal{N}(0, \sigma_i^2)$, with $\{\sigma_i^2\}$ such that $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$. ...
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### Rate of convergence for martingales, “merging of opinions” results

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be probability measures on this space. Let $(\mathcal{F}_{n})_{n \in \mathbb{N}}$ be a filtration on $\Omega$. Assuming that the ...
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### Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
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### What at the chances of getting 20 heads on a row if tossed 100 million times? [duplicate]

I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such ...
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### Example of set, finite outer measure, subsets, where outer measure does not converge

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ ...
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### Can we use a symmetry argument instead of integration in BASIC probability?

Suppose $H$ is a random variable with pdf $f_H(h)$. Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y} = f_H(x) f_H(y)$$ Prove $$P(X \ge Y) = 1/2$$ Is it possible to ...
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### Finding best strategies in a problem about traffic lights

A problem came up in a course of a conversation with my friend. Suppose we have a street with several traffic lights, placed equidistantly one from another. Time of them being green $t_g$ and time of ...
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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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### Probability of $|H-T|$ in 10,000 coin tosses

If a fair coin is thrown $10,000$ times. Using the binomial convergence to normal,find $P|H-T|\le 80$ My intuition say that mean is 0.But I am not able to proceed further.
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### Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
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### $\mathbb{E}[|X|^n] < +\infty \implies \mathbb{E}[|X|^k] < +\infty, k \leq n$

Show that if $\mathbb{E}[|X|^n] < +\infty$, then $\mathbb{E}[|X|^k] < +\infty, \forall k \leq n$. I guess I have to apply Hölder Inequality, but I was not able to find out what $p$ and $q$ are ...
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### Exponential martingale, Lévy-process and stopping times, definition quesiton.

I feel there is some ambiguity for the definition of the exponential martingale for a levy process which I do not understand. For a Lévy process it can be shown that $E[e^{iuX_t}]=e^{t\eta(u)}$, ...
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### Transformation of density and $W=(X+Y+Z)^2$

I want to solve this exercise with the transformation formula, what did I do wrong in my solution?: Let $X,Y,Z$ be independent random variables with uniform distribution on [0,1]. What's the ...