All Questions
Tagged with sequences-and-series probability
930
questions
2
votes
2
answers
60
views
Optimal strategy for uniform distribution probability game
There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
- 21
5
votes
5
answers
226
views
Closed formula for probability of n-digit numbers containing three consecutive sixes
I'm trying to find a closed formula $f(n)$ for the probability of choosing a number with $n$ digits that contains at least three consecutive sixes. Ideally, the formula should not depend on $f(n-1)$. ...
0
votes
0
answers
31
views
Arrangements of fixed k-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 159
2
votes
0
answers
65
views
Long term probability of going bankrupt?
Suppose there is a coin flipping game where you start with 5 dollars. At each turn, there is a $p$ probability of winning 1 dollar and a $1-p$ probability of losing 1 dollar. The game ends at 0 ...
- 785
1
vote
1
answer
46
views
What is the probability that 4 random bytes all increase in value?
From a randomness test @ https://www.vmpcfunction.com/c6.htm (2/3rds down the page):-
Four-Byte-Run-Up Test
The sequence is divided into four-byte segments. If the first byte of the segment is less ...
- 529
0
votes
1
answer
32
views
Simple ergodic convergence proof for iid
This is just a simple question from a problem sheet. Consider a sequence of independent identically distributed random variables $Y_0,Y_1,\dots$. Let $f$ be a function such that $\mathbb{E}|f(Y_0)|^2 &...
4
votes
2
answers
265
views
Convergence of random sequence$X_n$ specified by $X_n\leq aX_{n-1}$ with probability $1$ and $X_n\leq bX_{n-1}$ with probability $p$
I have a sequence of strictly positive numbers, $x_n$, and suppose I know deterministically $x_1=\alpha$. But $x_n\leq ax_{n-1}$ with probability $1$ and $x_n\leq bx_{n-1}$ with probability $p$. Here, ...
- 151
2
votes
0
answers
54
views
Convergence of a series for a triangular array of random variables
Inspired by my previous post (in which I believe there is an error in the answer I gave) I am interested in the following:
Suppose $\{X_{n,i}\}$ is a triangular array of row-wise iid random variables ...
- 188
1
vote
1
answer
65
views
Convergence rate governed by a random variable
Consider a sequence of nonnegative numbers, $\{x_n\}_{n\in \mathbb{N}}$, and a sequence of iid discrete random variable $\{Z_n\}_{n\in \mathbb{N}}$, where $Pr[Z_n=1]=p$, $Pr[Z_n=1/2]=1-p$, with $w\in (...
- 151
1
vote
1
answer
44
views
If a random variable converges to zero in probability what can we say about its almost sure boundedness?
First let me start with definitions that I will be using in the question.
A sequence of random variables $X_n(\omega)$ converges to zero in probability if for any $\epsilon>0$, and any $\delta>...
- 151
1
vote
1
answer
67
views
How to prove $\forall p\in\left(\frac12,1\right)\quad\sum_{k=0}^\infty \left( \frac1{k+1} \binom{2k}k p^k (1-p)^k \right) \le \frac1p$? [duplicate]
How to prove $\forall m\ge0\quad\forall p\in\left(\frac12,1\right)\quad\sum_{k=0}^m \left( \frac1{k+1} \binom{2k}k p^k (1-p)^k \right) \le \frac1p$?
Since this has to be true for all $m$ then it's ...
- 11
4
votes
1
answer
108
views
Evaluating $\lim_{n\to\infty}\sum_{i=1}^n \exp\left(\frac{-|x-X_i|^2}{2(\sigma/n)^2}\right)$
Let $X_1,...,X_n\stackrel{iid}{\sim} \mu$ where has a density with respect to the Lebesgue measure on $\mathbb{R}$: $\mu(dx)=\rho(x)dx$. For every $x$ show$$
\lim_{n\to\infty}\sum_{i=1}^n \exp\left(\...
- 188
0
votes
1
answer
23
views
Does the deterministic sequence dominated by a random sequence converges to zero in deterministic sense if the random sequence converges a.s.
I am new to the concepts of different types of convergence of random sequences. Suppose $\{a_k\}_{k\in\mathbb{N}}$ is a deterministic sequence. Let $\{X_k(\omega)\}_{k\in\mathbb{N}}$ be a random ...
- 151
1
vote
1
answer
36
views
Show that the maximum of negative binomial cannot converge to a non-degenerate distribution
Let $X \sim NegBin(m,p)$, where $m$ is a positive integer and $p \in (0,1)$.
$X$ is a random variable with $P(X=n) = {n+m-1 \choose n}(1-p)^n p^m$ for $n \geq 0$.
I want to show that as $n \to \infty$,...
- 1,710
3
votes
3
answers
95
views
Find Variance of Geometric Random Variable Using Law of Total Expectation
I'm trying to compute variance of geometric RV $X$ with parameter p. I would like to use the Law of Total Expectation. RV $Y$ represents the first trial, which is either success with probability $p$ ...
- 59