# Questions tagged [bernoulli-trial]

The bernoulli-trial tag has no usage guidance.

19
questions

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### Distribution of a two-part sampling process

I have a known distribution $f(x)$ (in fact, I can safely assume that $f(x)$ is the Maxwell-Boltzmann distribution, i.e. $f(x)\propto x^2 \exp(-x^2)$). I take $N$ samples from the distribution, but am ...

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### How to prove this "obvious" result from Bernoulli trials / partial Pascal series?

The main problem setup - very brief
I need to prove that the following two equations have a solution with $CAP=v$.
$$0=
( CAP - C ) \sum_{j=1}^J j N \cdot q^{j-1} (1-q)^{N -1-j} \binom{N}{j} ...

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### $X-Y$, where $X$ and $Y$ are sums of Bernoulli random variables

Let $X = x_1 + x_2 + \ldots + x_n$ and $Y = y_1 + y_2 + \ldots + y_n$, where each $x_i$ is an independent Bernoulli random variable with success probability $p_i$ and each $y_i$ is a Bernoulli random ...

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### How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?

Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...

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### Can the sum of identically distributed dependent Bernoulli trials be binomially distributed?

If you have $n$ identically distributed Bernoulli trials whose sum is binomially distributed random variable, does it then follow that the $n$ Bernoulli trials are independent?

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### $Pr(A>B)$, where $A$ and $B$ are sum of Bernoullies

Let $X= x_1 + x_2 + \ldots + x_m$, $Y=y_1 + y_2 + y_3 + \ldots + y_n$, and $Y' = y'_1 + y'_2 + \ldots + y'_n$, where
Each $x_i$ is a Bernoulli variable which takes value $1$ with probability $p_i>...

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### Probability distribution of Bernoulli trial of independent events with arithmetic progression probability [closed]

If $n$ tests are independent of each other, and the probability of an event in each test form an arithmetic progression, that is, the probability of the event in the first test is $p$, and in the ...

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346
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### Deriving the joint distribution of multivariate normal transformed into Bernoulli

Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......

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### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...

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### Cramér-Rao bound for randomized estimator

As is well known, the Cramér-Rao bound (or information inequality) sets a lower bound on the variance of estimators of a parameter.
Consider the case when the parameter is a scalar, the estimator is ...

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### Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...

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### Another name for coin-flipping polynomials

In his paper Functions arising by coin flipping (section 4), Johan Wästlund coined the term "coin-flipping polynomial" for polynomials that arise in connection with observing a finite number of coin ...

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### Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...

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### a function of Bernoulli variables?

Let $X_1,X_2,...,X_n$ be a fixed number of Bernoulli random variables. My problem is to find a distribution for $Y$ such that for some function $f$, we have $Y=f(X_1,X_2,...,X_n)$. There are two ...

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489
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### Rigorous proof of the duality of Coupon collector's problem and Occupancy problem

We have $k$ different types of coupons (with replacement).If we collect at least $l$ different coupons, we win a prize. We can only afford to collect $m$ coupons.
Let's say we take all those $m$ ...

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### Mean occurrences of letters in complete strings given by a Bernoulli scheme

Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...

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### Generating Bernoulli Correlated Random Variables with Space Decaying Correlations

Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a ...

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1
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### References for Poisson and Lexis trials

I have been trying to find more information on Poisson and Lexis trials (generalizations of Bernoulli trials), but I have failed to find anything outside of MathWorld (I went through a number of ...

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### When do binomial distributions occur?

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...