Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [bernoulli-process]

46 questions
60
votes
13 answers

Does 10 heads in a row increase the chance of the next toss being a tail?

I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of probability and/or statistical jargon is tossed around…
user68492
  • 601
  • 1
  • 6
  • 3
7
votes
0 answers

Is average stopping time a continuous function of Bernoulli parameter?

Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case that $\mathrm E[N]$ is a continuous function of…
Luis Mendo
  • 815
  • 7
  • 20
7
votes
1 answer

Independent Bernoulli trials vs markov chain

Original Question Suppose we have a sequence of Bernoulli trials $X_1, X_2, \cdots X_T$ which are ordered in time and may or may not be independent. I am interested in understanding the probability of success.. The way I'm thinking about it, I have…
knrumsey
  • 5,943
  • 17
  • 40
5
votes
0 answers

What is a name of this Bernoulli-like process with dependent trials?

The process is defined similarly to the Bernoulli process composed of $n$ Bernoulli trials. The difference is that the trials are dependent, that is: $$ P(X_i = 1 | X_1, ..., X_{i-1}) = \frac{m -\sum_{j = 1}^{i - 1} X_j}{m} p , $$ where m is a…
abukaj
  • 353
  • 1
  • 11
5
votes
1 answer

Number of Bernoulli trials to first success, with changing $p$

[A version of this question was previously posted by another user, but the OP deleted rather than edit the question into a more suitable form for routine textbook work. I am reposting in the hope of attracting an answer that was outlined…
Glen_b
  • 257,508
  • 32
  • 553
  • 939
5
votes
1 answer

Maximum likelihood estimation of a Poisson binomial distribution

According to Wikipedia, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed In other words, the Bernoulli trials have different…
anonymous
  • 51
  • 1
5
votes
1 answer

Age and residual life time of the Poisson process

Original Question Let $N(t)$ be a Poisson process with intensity $\lambda$. Let $T_10$, define the $age$ random variable to be $A_t := t-T_{N(t)} $ the $residual$ $life$ $time$ random…
Ye Tian
  • 619
  • 6
  • 19
5
votes
1 answer

Number of trials necessary to demonstrate Bernoulli process doesn't have mean p

I have a Bernoulli process that purportedly has mean $x$ but I hypothesize that the process actually has mean $q$. How many trials are necessary to demonstrate (to some confidence $p$) that the actual mean $\bar{x}$ is $<= q$. We can assume with no…
Cirdec
  • 155
  • 4
3
votes
1 answer

Bernoulli process and two exponentials

Suppose that a very long Bernoulli process gives a sequence with possible values: $A$ with probability $p$, and $B$ with probability $1-p$. The expected fraction of contiguous sequences of length $k$ that only contain $A$s must be $p^k$. I made some…
user1420303
  • 527
  • 3
  • 8
3
votes
1 answer

confidence intervals for the Poisson process ($\lambda$) sampled with uncertainty

Say, I have a Poisson process which was measured $N$ times, and each measurement produced $k_i$ value. Also, $k_i$ are events that I have to detect and my detection probability is $p$. In fact, I detect $\widetilde{k}_i$ which correlates with $k_i$…
Gideon Kogan
  • 250
  • 1
  • 10
3
votes
1 answer

Flat "geometric distribution" by varying the probability of the Bernoulli trail

In a simulation I am working on, each day (time step) there is a chance that a condition changes (at which point it is stuck in the changed condition). Setting this probability to a fixed value (say 5%) gives a geometric distribution (or exponential…
goryh
  • 151
  • 1
  • 1
  • 6
3
votes
3 answers

How to calculate a confidence interval for a series of Bernoulli Trials?

I have to test if a event have a p probability of happening. I can run this event as much times I like (given it can be run by a computer). So I was searching a way to test if the probability of this event is in an acceptable range for my case. How…
Bruno Andreetto
  • 33
  • 6
3
votes
2 answers

Success of Bernoulli trials with different probabilities and without replacement

Assuming $n$ independent Bernoulli trials with different probabilities, the Poisson binomial distribution is the discrete probability distribution that describes the number of $X$ successes. A Hypergeometric distribution is the discrete probability…
user2597079
  • 77
  • 7
3
votes
0 answers

Convergence of a sequence of Binomial variable with changing probability

Consider a $t\in(0,1)$. Consider, for $\Delta>0$ the random variable $X_t^{(\Delta)}$ defined as $$ \mathbb{P}[X_t^{(\Delta)}=1]=\left(1-\lambda\,\Delta\right)^{\left\lfloor t/\Delta\right\rfloor},\quad…
AlmostSureUser
  • 213
  • 1
  • 9
2
votes
2 answers

Subsample of a random sample?

I am stuck with a very simple question, but I don't really understand sampling, so please help me. Assume that I perform Bernoulli sampling with parameter $q$ on data D, and obtain sample S1. Then on S1, I perform another Bernoulli sampling with…
Long Vehicle
  • 21
  • 3
1
2 3 4