All Questions
320
questions
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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 ...
- 593
0
votes
1
answer
17
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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 ...
- 593
2
votes
2
answers
71
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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 ${X_1,X_2,\dots}$ ...
- 1,011
1
vote
1
answer
58
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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 ...
- 1,011
2
votes
1
answer
92
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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
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3
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224
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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 $2$ ...
- 9,599
2
votes
1
answer
59
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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 ...
- 151
1
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0
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45
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Is $f$ a probability mass function for a random variable $X$ if and only if it complies with these two properties?
I've read in multiple books that a function $F$ is a cumulative distribution function for some random variable $X$ if and only if it satisfies three conditions:
$F$ is non-decreasing.
$F$ is ...
- 5,166
0
votes
0
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42
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Why is Bernstein's Inequality legal here?
This is a follow up question based on this post: Bounding rows of random matrices.
I can not see why we can use Bernstein's inequality as mentioned in the answer. In particular, it seems like the ...
- 593
1
vote
1
answer
52
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Expected Value Question 30 sided/20 sided die solution Issue
I recently saw a problem wherein player $A$ is given a $30$ sided die, player $B$ is given a 20 sided die, and they both simultaneously roll and whoever gets the higher outcome wins that dollar amount....
- 5,880
0
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1
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68
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Probabilistic Combinatorial problem - consecutive runs of $1$s in a string of $0$s and $1$s
Let $n$ values $0$ or $1$ be arranged around a circle, and for a given $k$ consider the number of runs of $k$ consecutive $1$s. Suppose the $n$ values are independent and each of them is equal to $1$ ...
user1289888
1
vote
2
answers
164
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Let $P(X_n=n)=1/n^a$ and zero otherwise. Let $Y_n=X_{n+1}\cdot X_n$. Find all $a$ such that $\liminf Y_n=0$ and $\limsup Y_n=\infty$ almost surely.
Remark: My attemps is WRONG as I fasly assumed that the $Y_n$ were independant. I ve corrected this in my own answer to this post. This answer is correct but not enough well wrotten, read the selected ...
- 719
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1
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42
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$X$ a r.v. verifying $P(X=x)=0$&$F_X$ its repartition fct. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Question:
Let $X$ a r.v. verifying $P(X=x)=0, \forall x \in \mathbb{R}$ and $F_X$ its repartition function. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Answer:
1- We write $0 \leq ...
- 719
2
votes
2
answers
87
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Indpendence inequality over interval: $\mathbb{P}[a<X<b]\leq\mathbb{P}[a<X]\mathbb{P}[X<b]$
I am trying to show the following:
$$\mathbb{P}[a<X<b]\leq\mathbb{P}[a<X]\mathbb{P}[X<b]$$
where $X$ is any random variable and $a,b$ are constants. I feel I am missing the obvious here, ...
- 135
3
votes
0
answers
48
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Random Walk Problem in Feller Probability Theory vol 2 Chapter XII page 425
Let $X_{i}$ be iid discrete random variable with distribution $P(X_{1}=-1)=q$ and $P(X_{1}=i)=f_{i}$ for all $i\geq 0$.
Then consider the random walk $S_{n}=\sum_{i=1}^{n}X_{i}$ and $S_{0}=0$. Let $\...
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