# 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.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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{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) \...
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### 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 $\exp(c^2t^2)$ ...
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### 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 $f$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\dfrac{\sum_{k=1}^{n}X_{k}}{\sum_{k=1}^{n}p_{k}}$ ...
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### 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?...
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### 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 ...