All Questions
Tagged with probability-theory solution-verification
843
questions
1
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0
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27
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Centered Subgaussian Variables have better Properties
I am trying to understand the following proof:
Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is ...
- 583
0
votes
1
answer
17
views
Understanding the proof for Properties of Subgaussian Variables
Here are the definitions, statements and the proof that I am stuck on:
I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds ...
- 583
0
votes
0
answers
14
views
Proving statement about cumulative distribution function
I want to prove the following statement:
Let F fulfill the properties of a cumulative distribution function. Define $$X^-(\omega) = \inf \{z \in \mathbb{R}: F(z) \geq \omega\} \quad X^+(\omega) =\inf ...
- 1
2
votes
2
answers
69
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A convergence property for iid sequence of Cauchy random variables
A real random variable ${X}$ is said to have a standard Cauchy distribution if it has the probability density function $\displaystyle {x \mapsto \frac{1}{\pi} \frac{1}{1+x^2}}$. If ${X_1,X_2,\dots}$ ...
- 971
1
vote
1
answer
58
views
Law of large number for non-integrable random variables
Let ${X_1,X_2,\dots}$ be iid copies of an unsigned random variable ${X}$ with infinite mean, and write ${S_n := X_1 + \dots + X_n}$. Show that ${S_n/n}$ diverges to infinity in probability, in the ...
- 971
-1
votes
1
answer
36
views
Sigma-algebra generated by $\sin(2\omega)$
If $\Omega = (0, 2\pi)$ and $X(\omega) = \sin(2\omega)$, is $\sigma(X) = \{S: x \in S \cap [0, \pi / 4] \Leftrightarrow \pi / 2 - x \in S \cap [\pi / 4, \pi / 2] \Leftrightarrow \pi + x \in S \cap [\...
0
votes
1
answer
51
views
Microstate interpretation of Shannon entropy
Let ${A}$ be a finite non-empty set of some cardinality ${|A|}$, and let ${X}$ be a random variable taking values in ${A}$. Define the Shannon entropy ${{\bf H}(X)}$ to be the quantity $\displaystyle ...
- 971
0
votes
0
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47
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Estimate the number of Cramer model primes
This is a sub-post of the problem Probabilistic Riemann hypothesis.
Consider a sequence of independent Bernoulli random variables $(X_n)_{n \geq 3}$ of parameters $1 / \log n$ (so $X_n = 1$ with ...
- 971
0
votes
1
answer
29
views
If $X = g(Y)$, then Is the sigma-algebra generated by $X$ a subset of the sigma-algebra generated by $Y$?
Let $g$ be a continuous function. Let random variables $X$ and $Y$ satisfy $X = g(Y)$. Do we always have $\sigma(X) \subset \sigma(Y)$?
I want to disprove this. My thought is that there must be a set $...
2
votes
1
answer
45
views
Sufficient Condition on Almost Surely Convergence
Let $f_n \in [0, 1]$ and suppose if we want to show
$$
\lim_{n \to \infty} f_n = 1
$$
almost surely, is it enough to show
$$
\lim_{n \to \infty} \mathbb{P}\{ f_n = 1 \} = 1?
$$
If not, what if we add ...
- 583
3
votes
1
answer
118
views
Exercise on Girsanov's theorem
currently I am trying to solve the following exercise.
Exercise: Let $b : \mathbb R → \mathbb R$ be Lipschitz, and let $t \mapsto X(t)$ be the unique strong solution of the 1-dimensional SDE given by
$...
2
votes
1
answer
92
views
Does $X_nY_n=\mathcal{o}_{p}(\beta_n)$ hold?
Let $X_n$ and $Y_n$ both be sequences of nonnegative random variables.Define $A_n:=\left\{\omega:X_{n}(\omega)>0\right\}.$ Suppose that $\lim_{n\rightarrow\infty}\mathbf{P}(A_n)=0,Y_n=\mathcal{O}_{...
- 137
1
vote
0
answers
23
views
Confusion about LLN for Empirical Processes/Measures
I have seen the following LLN result on empirical measures:
(Statement 1) Let $X, X_1, X_2, \cdots, X_n$ be i.i.d. random variables taking values in $[0, 1]$. Then
$$
\mathbb{E}\sup_{f \in \mathcal{F}...
- 583
-1
votes
3
answers
206
views
Two cards are drawn from a well shuffled pack of $52$ cards. Find the probability that one of them is a red card and the other is a queen.
Two cards are drawn from a well shuffled pack of $52$ cards. Find the
probability that one of them is a red card and the other is a queen.
My Attempt
The relevant cards are $26$ red cards and $2$ ...
- 9,569
2
votes
1
answer
57
views
Decaying inequality in expectation implies almost sure convergence to zero?
Is this claim true? The following is my attempt at the proof. I am unsure about the proof because I did not have to use the fact that $X_n\geq 0, \forall n\in \mathbb{N}$. Any feedback for ...
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