Questions tagged [bernoulli-distribution]

Use this tag in reference to questions concerning random variables following the Bernoulli distribution, including calculating quantities such as expectation, standard deviation, moments, as well as real world applications.

Filter by
Sorted by
Tagged with
1 vote
1 answer
44 views

Topological entropy of a Bernoulli Shift

I am approaching the world of entropy and I would like to have a few examples in mind. I know some chaotic systems that are topologically conjugated to the Bernoulli shift, so I would like to know the ...
Andrea Marino's user avatar
2 votes
2 answers
209 views

Is this a Bernoulli experiment?

In a volleyball tournament, two equally strong teams face each other. A match is considered won if one team wins three games. What is the probability that it will be over after three, four or five ...
bochner.martinelli's user avatar
4 votes
1 answer
54 views

Finding the limiting distribution of $T_{n}/S_{n}$ as n tends to infinity

Question Let $X_i \sim\left(i . i\right.$. $d$.) Bernoulli $\left(\frac{\lambda}{n}\right), n \geq \lambda \geq 0$. $Y_i \sim\left(i\right.$ i. d.) Poisson $\left(\frac{\lambda}{n}\right),\left\{X_i\...
Debu's user avatar
  • 63
0 votes
0 answers
31 views

Confidence interval 100 coin tosses, 52 heads?

I am really confused about the confidence interval for n=100 tosses and k=52 heads. If I estimate the probability $\hat{p}$: $\hat{p} = \frac{52}{100} = 0.52$ $Var(\hat{p}) = pq = 0.52 * 0.48 = 0.2496$...
DamianoDantuono's user avatar
-1 votes
1 answer
79 views

How to deduce that two random variables are equivalent?

For example, \begin{equation} Y_1= \begin{cases} x & \text{with prob. } p\\ 1-x & \text{with prob. } 1-p \end{cases} \end{equation} and \begin{equation} Y_2=\begin{cases} x & \text{with ...
Privatizer's user avatar
0 votes
0 answers
22 views

Mutual information between standard gaussian and bernoulli distribution

I have a random variable which is distribuited as a multinomial standard gaussian $z\sim N^d(0, I)$, and another r.v. that instead is distribuited as a bernoulli $\alpha \sim Bern(\pi)$. Is it ...
Francesco De Santis's user avatar
2 votes
0 answers
18 views

What is the PMF of the ratio of two dependent sums of weighted Bernoulli random variables?

Let $ X_i $ be a Bernoulli random variable of success probability $p$. Is there any closed-form expression of the probability mass function of the following quantity: $$ \frac{\sum_i a_i X_i + b}{\...
Kevin B.'s user avatar
1 vote
1 answer
29 views

Weak law of large numbers for distributed Bernoulli random variables in a particular case. [closed]

Let $X_1, X_2, ...$ be a sequence of independent and identically distributed random variables, such that $X_i\sim Bern(p)$. Now, let $Y_{1,n}, Y_{2,n}, ...$ be random variables, such that $Y_{i,n}\sim ...
Helder Alves Arruda's user avatar
0 votes
0 answers
20 views

Linear system with multivatiate bernoulli elements

I have the following system $$ \mathbf M \left(\alpha_1,\alpha_2,...,\alpha_n \right) \vec v = \vec b $$ $$ v_{out} = \vec c^T \vec v $$ where $\mathbf M \left(\alpha_1,\alpha_2,...,\alpha_n \right)$ ...
Torben's user avatar
  • 3
0 votes
1 answer
76 views

expected value of the product of two Bernoulli variables [closed]

I'm stuck on proving that the product of two Bernoulli variables has expected value equal to the difference between the probability that the variables are the same and the probability that the ...
bravoralph's user avatar
1 vote
3 answers
91 views

Probability of A and B and C occurring exactly once

A spinner has 6 equal sectors; A,B,C,D,E,F. All have equal probability of 1/6. Find the probability of A and B and C exactly once if the spinner is spined 10 times. Also similarly find probability of ...
vikram chaudhari's user avatar
0 votes
1 answer
29 views

Variance of product of two random variables where one is a Bernoulli whose probability is a function of the first variable.

I have the following problem. I have a Gaussian variable $x$ with mean $\mu$ and variance $\sigma$ and I construct a certain function $0 < f(x) < 1$. This function value becomes then the ...
franyx's user avatar
  • 1
0 votes
2 answers
106 views

Entropy of XOR between two Bernoulli variables

I got: $X$ and $Y$ independent Bernoulli random variables, and $Z = X \oplus Y$ (note that $Z$ is also Bernoulli variable), entropy values are: $$H(Z)=−\left[(1−p)(1−q)+pq\right]\log\left[(1−p)(1−q)+...
Adi .k.'s user avatar
2 votes
0 answers
87 views

A seemingly obvious inequality: the restricted multinomial probability $\geq$ the product of binomial probability

Given a normalized real vector $\theta \in \mathbb{R}^n$ (its elements sum up to $1$, i.e., $||\theta||_1 = 1$), an integer vector $x \in \mathbb{Z}_{\geq 0}^n$ and an integer $S \geq \sum\limits_{i=...
Geek's user avatar
  • 23
1 vote
1 answer
65 views

Binomial distribution with variable parameters

This question is motivated by a problem on the harmonic measure. I also asked this on Mathoverflow, copying it here, and hope that cross-posting here is not a problem. In each trial of a sequence of ...
Aritro Pathak's user avatar
0 votes
1 answer
113 views

Is the distribution of sample mean of Bernoulli random variable a Binomial distribution?

As written in the title, is the distribution of sample mean of Bernoulli random variable a Binomial distribution? And I was taught that we can approximate Binomial distribution to normal distribution ...
비선형편미분방정식's user avatar
2 votes
0 answers
176 views

Minimize the joint entropy of Bernoulli variables with given marginal distributions

Given a list of Bernoulli variables $X_1, X_2, \ldots, X_n$ with fixed marginal distributions (i.e., success probabilities) $p_i = \Pr[X_i = 1], \forall i \in [n]$. Question: How can we minimize the ...
Vezen BU's user avatar
  • 1,981
0 votes
0 answers
11 views

Is the greedy projection the best (entropy-minimizing) projection?

Let $Y,X_1,X_2$ be binary random variables. I'm training a naive binary tree classifier and I need to find the best binary projection $f: \{0,1\}^2 \to \{0,1\}$ such that $H(Y|f(X_1,X_2))$ is minimal. ...
user35443's user avatar
  • 373
0 votes
0 answers
12 views

What distribution do you get by repeatedly applying beta distribution?

So I know repeatedly adding 0 mean gaussians gives another 0 mean gaussian with the variances adding. I wanted to get a better understanding of if there's an analogue to this for variables ranging ...
Chinmay The Math Guy's user avatar
0 votes
0 answers
30 views

Question regarding independent (Bernoulli) trials: rolling a die 10 times and defining success as rolling a 1 or a 6.

I just want to make sure that the scenario I'm describing is indeed a Bernoulli trial type of experiment. Here is the set up and query regarding the set up: A six-sided die is rolled 10 times. What is ...
Mariusz Popieluch's user avatar
0 votes
1 answer
33 views

Given bivariate bernoulli with an integral as a parameter prove that are marginally identically distributed and correlation is positive

So this is a question from a past exam. The joint density function is \begin{equation} \begin{aligned} P\left(X_1=x_1, X_2=x_2\right) & =I_{\{0,1\}}\left(x_1\right) I_{\{0,1\}}\left(x_2\right) \...
Guilherme Marthe's user avatar
2 votes
0 answers
130 views

subexponential constant of Bernoulli

We say that a random variable $X$ is subexponential with constant $c$ if $$ \mathbb{E}[\exp(t(X-p))] \le \exp(c^2t^2), \qquad |t|\le 1/c. $$ If instead the MGF were upper-bounded by $\exp(c^2t^2)$ ...
Aryeh's user avatar
  • 440
0 votes
0 answers
18 views

Understanding the sum of Bernoulli variables, and functions defined on such sums.

Let $y_1,y_2$ be independent Bernoulli variables, and let $f(y_i)=ay_i+1$ be a function defined on both Bernoulli variables. Also, let $B_i(f(y_i))=ay_i$ be an operator acting on the function $f$. ...
matilda's user avatar
  • 169
2 votes
0 answers
58 views

The expectation of a Bernoulli divided by an associated sum of bernoullis

For $i=1,\dots, k$, let $X_i \sim \mathrm{Bernoulli}(p_i)$ be independent such that $\sum_{i=1}^k p_i > 0$. How do I evaluate the following expectation: $$\mathbb{E} \left( \frac{X_i}{\sum_{j=1}^k ...
user551504's user avatar
0 votes
0 answers
20 views

How to Update a Beta Prior Based on Observations from a Product of Two Independent Bernoulli Variables

I'm working on a problem involving Bayesian updating with a Beta prior, but the data I observe comes from a slightly complex source. Let $X \sim \text{Bernoulli}(p)$ and $Y \sim \text{Bernoulli}(q)$, ...
Parchment2382's user avatar
5 votes
1 answer
127 views

Stochastic differential equation with Bernoulli random process as a solution

Let us assume that $T$ is an absolutely continuous random variable such that $T>0$ almost surely and $\mathbb{E} e^T < \infty$. Let us define $$ X_t = \mathbb{I}_{ [ t <T ] }, \ t \geq 0 .$$ ...
user avatar
3 votes
2 answers
136 views

Bernoulli Bootstrapping and the Beta Distribution

My understanding of the bootstrap is that it gives us a method to understand the distribution of an estimator applied to a dataset. I've read statements of the form "bootstrapping relies on the ...
Steve's user avatar
  • 83
3 votes
2 answers
213 views

What is the expected time at which a success $S$ followed by a failure $F$, occurs for the first time in a sequence of Bernoulli trials?

We have $P(S) = a$ and $P(F) = 1-a$. I want to find the expected time at which $SF$ occurs for the first time. I'm following a method given in my lectures but not getting the right answer so I must be ...
spooleey's user avatar
  • 436
0 votes
1 answer
48 views

$\mathbb P(A > B)$ when $A$ and $B$ are independent random variable with continuous Bernoulli Distribution.

Suppose $A$ and $B$ are independent random variables with continuous Bernoulli distribution: \begin{align} f_A(x)&=\lambda_a^x(1-\lambda_a)^{1-x}\frac{2 \tanh ^{-1}\left(1-2 \lambda_a\right)}{1-2 \...
huangbiubiu's user avatar
0 votes
1 answer
50 views

First ocurrence of a pattern

Let $X_0, X_1 , \dots$ be independent identically distributed Bernoulli random variables such that $$ \mathbb{P} [ X_k = 0 ] = \mathbb{P} [ X_k = 1] = 1/2, \ k \geq 0 $$ Let us define $$ \tau_{110} = \...
user avatar
0 votes
1 answer
66 views

Probability of infinit sum of Bernoulli random variables with different parameter to

Suppose there is an urn with $n\in\mathbb{N}$ balls, of which one is marked. If we draw the marked ball from the urn it was a 'good' draw, after each draw we put the drawn ball back into the urn with ...
sombrero's user avatar
2 votes
2 answers
120 views

What is the absolute expected value of the sum of a distribution made up of Bernoulli and Normal distributions?

$$X \sim \mathcal N(\mu, \sigma^2)$$ $$Y \sim Bern(p)$$ $$Z = XY$$ I then have that the pdf of $Z$ is: $$f(z) = (p-1)\delta(z) + p g(z)$$ where $g(z)$ is the pdf of the normal distribution. I then ...
Filippo's user avatar
  • 21
1 vote
1 answer
136 views

Probability of $x$ trials given $k$ successes

I am looking for a probability distribution that calculates the probability of $x$ total Bernoulli trials given fixed $k$ successes. I have looked into negative binomial distribution: $$ P(X=x) = \...
Marcus Chiu's user avatar
1 vote
1 answer
367 views

Why is a sequence of random variables not a markov chain?

I have a sequence of random variables: Y1, Y2, Y3, .... where Y1 and Y2 are both i.i.d. Bernoulli(0.5), and for all j >= 3, the following holds: if min(Yj-1, Yj-2) = 1, then Yj is Bernoulli(2/3), ...
Shatarupa18's user avatar
3 votes
0 answers
77 views

Expected value of the parameter $P$ of a Bernoulli having observed $s$ successes over $t$ trials, when $P$ is uniform on $[a,b]$ with $0\le a<b\le 1$

Using the relation between Beta and Gamma functions $$B(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$ where $$B(m,n) = \int_{[0,1]} p^{m-1}(1-p)^{n-1}\operatorname{d}p$$ and the fact that for any $n ...
Bob's user avatar
  • 5,725
1 vote
2 answers
59 views

Proving linearity of the expected value of two (possibly dependent) Bernoulli variables

I'm currently working on proving that the expected value operator is linear, and felt quite content with my work until recently. However, I was recently made aware that my proof is somewhat lacking, ...
hgrenersen's user avatar
0 votes
2 answers
117 views

Why is the Bernoulli distribution not discrete?

Problem In any basic probability course, we learn that the Bernoulli distribution is a discrete probability distribution. However, when using measure theory, it seems that it is actually not. ...
Physics_Student's user avatar
0 votes
1 answer
87 views

Expected value of the inverse of a random variable similar to a Binomial but where $0$ is not supported

Let me start by defining the random variable. I have $n$ units. I run a (Bernoulli) experiment where a unit is assigned to group $U$ with probability $q$ or to group $U'$ with probability $1-q$. Let $...
lovatic27's user avatar
1 vote
0 answers
45 views

Why are square Bernoulli matrices invertible with high probability?

Given a matrix $A\in\mathbb{R}^{m\times n}$ with entries of $A$ being sampled i.i.d. from $\text{Bernoulli}(\alpha)$, where $\alpha\in(0,1)$ is a fixed constant. This paper (2nd sentence below ...
Resu's user avatar
  • 764
2 votes
0 answers
40 views

Sampling of independent Bernoulli variables with fixed cardinality

Suppose we have $n$ probabilities $p_1, p_2, \ldots, p_n \in [0, 1]$, and let $A_1, A_2, \ldots, A_n \in \{0, 1\}$ be corresponding independent Bernoulli events, where $A_i = 1$ with probability $p_i$....
Vezen BU's user avatar
  • 1,981
2 votes
1 answer
50 views

The variance of the average of Bernoulli random variables

Suppose to have two triangular arrays of Bernoulli random variables, that is, for all $n\in\mathbb{N}$, the collections of random variables $\{A_{j,n}| j=1,...,n\}$ and $\{B_{j,n}| j=1,...,n\}$ are ...
AlmostSureUser's user avatar
1 vote
0 answers
38 views

Central limit theorem for sum of dependent Bernoulli random variables in multivariate hypergeometric setting

I am struggling to find a suitable Central Limit Theorem (CLT) for dependent variables for the following example: We have $K$ bins. In each bin $B_i$, $i=1,...,K$, we have $N_i$ balls, with $N=\sum_{i=...
Nepel3's user avatar
  • 11
2 votes
1 answer
148 views

Are Bernoulli distributions log-concave?

Question: I am aware that the common continuous distributions (like Gaussian, Uniform, Gamma) are log-concave. I am wondering if Bernoulli distributions (a discrete distribution) is log-concave? If so,...
Resu's user avatar
  • 764
0 votes
3 answers
62 views

How to calculate conditional probability in Bernoulli trial?

Let a fair die is thrown $10$ times. Let $X_i$ denotes the number of times the digit $i$ appears. What would be the probability $P(X_2=2|X_1=2)$ ? By formula $P(X_2=2|X_1=2)=\frac{P(X_1=2~\cap~ X_2=2)}...
MAS's user avatar
  • 10.7k
0 votes
0 answers
59 views

Intuitive utility of the Jeffreys prior, eg. in Bernoulli trials

I understand the computation the Jeffreys prior, and also its historical motivation. I (somewhat) understand the theoretical desirability of a "prior-construction principle/method" that is ...
cambridgecircus's user avatar
0 votes
1 answer
84 views

Rosenthal First Look at Rigorous Probability Theory Uniform Construction from Bernoulli

I am reading the mentioned book by Rosenthal, in particular pg 74 where a uniform is constructed from an infinite sum of i.i.d. Bernoulli, i.e., $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ with $$U=\sum_{k=1}^{\...
nvm's user avatar
  • 1,296
1 vote
2 answers
60 views

Variance of "grouped" summation of Bernoulli variables

Given $n$ objects (WLOG, $[n] = \{1, 2, 3, \ldots, n\}$) and two probabilities $p, q \in (0, 1)$. Starting from $X = [n]$ and $Y = \emptyset$, let us consider the following repeated process: Sample ...
Vezen BU's user avatar
  • 1,981
4 votes
1 answer
56 views

Suppose $X_{n}\sim bern(p_{n})$ and $\sum_{n}p_{n}<\infty$. Then $\frac{\sum_{k=1}^{n}X_{k}}{\sum_{n}p_{n}}$ does not converge to $1$ in probability

This may seem like a weird question. Suppose $X_{n}\sim \operatorname{bern}(p_{n})$, independent and $\sum_{k=1}^{\infty}p_{k}<\infty$. Then does $\dfrac{\sum_{k=1}^{n}X_{k}}{\sum_{k=1}^{n}p_{k}}$ ...
Dovahkiin's user avatar
  • 1,183
0 votes
0 answers
52 views

Joint distribution of bivariate normal and bernoulli

Let (X,Y) be a bivariate normal and Z follow Bernoulli distribution and be independent of (X,y). Its mean $(X,Y)\sim N(\mu,\sigma I)$ and $Z \sim Ber(p)$. How can I find the joint distribution of them?...
Long Tuấn's user avatar
2 votes
0 answers
49 views

Entropy of a Bernoulli random variable with logit probability.

I need to compute the expected entropy of a Bernoulli random variable whose probability of success is given by $\frac{1}{1+\exp(-x)}$ where $x$ is normally distributed. This leaves me stuck with ...
user449277's user avatar