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The Bernoulli distribution is a discrete distribution parametrized by a single "success" probability. It is a special case of the binomial distribution.

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Bayesian inference - iterative updating with Bernoulli distribution (solved)

Suppose I pull samples from a Bernoulli distribution $\mathcal{B}(\theta)$ I don't know the value of $\theta$, but in my case I know that $\theta$ can only have 11 discrete values, $\theta \in \{0....
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Likelihood Ratio Given Conditionals

For each person, I have at least one report indicating whether they have a disease or not. I have the actual disease status of a decent chunk of this population, and I'd like to be able to predict ...
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Finding MLE of $p$ where $X_1\sim\text{Bernoulli}(p)$ and $X_2\sim\text{Bernoulli}(3p)$

Let $X_1\sim\text{Bernoulli}(p)$ and $X_2\sim\text{Bernoulli}(3p)$ be independent Bernoulli random variables where $p\in[0,1/3]$. Derive the MLE of $p$. We have that $$L(p\mid \vec{x})=p^{x_1}(1-...
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Does mixture of sigmoids make sense given the theories about mixture of bernoullis?

Mixture of bernoullis is the combination of bernoulli distributions, which can be illustrated by the sampling process of K bags of D coins, here is a quick tutorial about it https://cedar.buffalo.edu/~...
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66 views

Maximum likelihood estimator for Bernoulli parameter based on standard normal

$X_i \sim Normal(\psi,1), \ \ i = 1, ..., n$ $Y_i = 1$ if $X_i \ge 0.$ $Y_i = 0$ if $X_i < 0.$ Let $\theta = P(Y_i = 1)$. What is the MLE of $\theta$? I know how to find the MLE of a Bernoulli ...
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Bernoulli / binomial trials for a process with variable probability of success

One of the conditions for a Bernoulli trial (and by extension binomial proportion confidence intervals) is that the probability of success is the same every time the experiment is conducted. In the ...
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Comparing 2 Bayesian Models with different structure

I'm a bit new to Bayesian statistics so please bear with me if this question is trivial. Let's say I have $100$ observations for $2$ Bernoulli variables $X$ and $Y$. I notice that they have the ...
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Detecting change in p of a Bernoulli process

A machine outputs either a 0 or a 1 each second. We denote this output at time $t$ as $b_t$. The probability that it outputs 1 is $p_t$ at time $t$. How do we go about studying the change in $p_t$ in $...
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Tutorial question on min number of sample size for confidence interval

I'm stuck with this question from my tutorial (and there is no worked solution), and I can't seem to get the correct answer of 411. There were 904 new Subway Restaurants franchises opened during 2002....
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26 views

Joint Posterior Distribution

I have 4 groups, each has a probability of developing gout (Bernoulli distribution), with a total of 400 individuals. I am confused how to derive and present the joint posterior distribution for each ...
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10 Trials: Each with 2% Success Rate, what is the Probability One of the Trials will be successful?

I'm looking for chance of success when within a number of trials with each trial having success rate x I learned that formula in highschool stats but I've since forgotten it. Oh what a fool am I! My ...
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How can we model a continuous coin tossing system to predict next tossing result and having a variable bias in the coin [duplicate]

Let's assume we have an unfair coin and a machine that toss it continuously. We counted the number of tandem heads. Whenever it's head we count 1, if it's head again, counter goes to two and so on. ...
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Simulating Bernoulli sample mean confidence interval in python

I'm working through exercises in my statistics textbook, and I'm getting a result I don't understand that I don't know if it's a programming problem or an understanding statistics problem. I'm trying ...
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774 views

With what probability one coin is better than the other?

Let's say we have two biased coins C1 and C2 both having different probability of turning head. We toss ...
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Detecting outliers in binary data using Mahalanobis distance

I have a binary vector $X_i$, $i=1...N$ of independent Bernoulli variables with parameters $p_i, \mu_i = p_i, \sigma_i^2 = p_i(1-p_i)$ (which is known) and I'm looking for some sort of tail bound to ...
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What is the distribution of a sum of identically distributed Bernoulli random varibles if each pair has the same correlation?

What is the distribution of a sum of $n$ Bernoulli random variables, each having success probability $p$, where each pair is correlated with correlation coefficient $\rho$? $$Y = \sum_{i=1}^n X_i$$ $$...
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Goodness of fit test for observed samples of binary strings

Consider a process that produces binary strings of varying length $n$. A typical sample would include $n\ $ I Number of strings $1\ $ I $\ 3,000,000$ $2\ $ I $\ 800,000$ $3\ $ I $\ 350,...
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Sum of independent binomially distributed variables (with different p's)?

The sum of independent variables each following binomial distributions $B(N_i,p_i)$ is also binomial if all $p_i = p$ are equal (in this case the sum follows $B(\sum_i N_i, p)$. If the $p_i$ are ...
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Upper bound for the probability $P\left[\left|\frac{Y_n}{n}-p^2\right|>\varepsilon\right]$

Let $X_1,X_2,\cdots,X_{n+1}$ be independent random variables with $$P(X_i=1)=p=1-P(X_i=0)\quad\text{ for all }i$$ Define $Y_i$ to be the number of $i$'s such that $X_i=X_{i+1}=1\,,\quad i=1,2,\...
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Tossing coin and classical ML estimate

I'm reading Bishop's Pattern recognition and came across with the next on the p.23: Suppose, for instance, that a fair-looking coin is tossed three times and lands heads each time. A classical ...
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Verification of sufficiency of a linear combination of the sample $(X_i)_{i\ge1}$ where $X_i\stackrel{\text{i.i.d}}\sim\text{Ber}(\theta)$

This question is in regards to this post where it asks if a certain statistic is sufficient for the parameter or not. My query is specifically with this problem: Let $X_1,X_2,X_3$ be i.i.d ...
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Difference between Empirical distribution and Bernoulli distribution

I've been studying binary cross entropy error for binary classification weight optimization. From my knowledge, Cross entropy itself quantifies divergence between two probability distributions with ...
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Monte Carlo simulation--Have I applied Bernoulli distribution properly? [closed]

I am trying to run a monte carlo simulation and I just wanted to make sure that I set it up properly. A salesman visits 100 different homes. Someone answers the door 80% of the time. Of that ...
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Why the variance of a proportion using multiple survey questions is the same as the proportion of only one survey question?

I am measuring the proportion of a sample that gets all successes in 10 different questions of a survey. For example, one question is "Do you smoke?" and a success for me is "No". Another question ...
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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 ...
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Can Bernoulli random variables be used to approximate more than just the normal distribution?

Most statistics students are familiar with the normal approximation of the binomial distribution. And since binomial distributions are created from sums of Bernoulli random variables, it would follow ...
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Two Bernoulli distribution (test hypothesis of p of biased coins from a sample)

I'm simplifying a research question that I have at work. Assuming I have 2 coins each with a different probability of head, let's call heads a success (p). Those ...
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Why is this a Bernoulli distribution?

In the paper I am reading, I come across $$ q(s) \propto \left( \frac{b}{c} \right)^{s}\quad s=\{0,1\}, \quad(1) $$ and the author says this is a Bernoulli distribution. ($b>0$ and $c>0$) I ...
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Better skill test for RPGs - Conditional probability given 2 independent parameters

I am trying to find a better way (theoretically, not practically speaking) to roll the dice for a skill test in RPGs. In the d20 system, the Game Master choose a Difficulty Level for the skill test, ...
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dbinom for Bernoulli trials

I have this question: "There are a 100 families each with 5 children. Given that the null probability of having a boy is $p=0.5$, what is the probability of a family having 0,1,2,3,4,5 boys" We have ...
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Convert exponential to Bernoulli

If I have a binary variable x, with distribution p(x) = exp{Cx}, how do I put this into the canonical Bernoulli form so as to get the probability p that x=1 that I ...
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power calculation for two stage binomial type model

This is a power-type calculation for a Bernoulli/binomial question in two stages. Suppose you are planning an experiment which starts with a test for an event on $N$ experimental units. The event ...
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Entropy of the beta-binomial compound distribution

I have a generative process as follows: $$ x \mid \alpha \sim \textsf{Beta}\left (\alpha,\beta \right) \\ y \mid x \sim \textsf{Bernoulli}(x). $$ How does one go about calculating the Entropy of ...
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Clarification: Bernoulli random variable with uniform distribution

Let $Z$ be a random variable which takes the value 1 when $U \le \frac 14$, $0$ otherwise, where $U$ ~ $\text{Uniform}(0,1)$. So $Z$ is a Bernoulli random variable with PMF $$p_Z(z) = \begin{cases} p,...
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Show that Bernoulli has Poisson distribution with $p\lambda$ if $\xi=k$

I have the following problem set at hand: The random variable $\xi$ has Poisson distribution with the parameter $\lambda$. If $\xi=k$ we perform $k$ Bernoulli trials with the probability of success $...
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Entropy of a matrix with Bernoulli distributed (binary entries) row-vectors

The entropy $H[x]$ of a Bernoulli distributed binary random variable $x$ is given by : $$ H[x]=−θlnθ−(1−θ)ln(1−θ) $$ where $$ p(x=1∣θ)= \theta \\ p(x=0∣θ)=1−θ $$ Now, suppose I have a vector as so: ...
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Saturated model for a Bernoulli response

I have a have a saturated GLM for a Bernoulli response. Let $Y_i \backsim \text{ Ber}(\pi_i)$. The saturated model yields $\pi_i^{(s)}=y_i$, where $y_i$ is the observed value, which implies that $$\...
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Bernoulli variable expected value and variance

Problem: Let $X$ be a Bernoulli random variable with $P(X=1)=p$. What are $E(5X^{10}+4^X)$ and $\text{Var}(5X^{10}+4^X)$? My work so far: $E(5X^{10}+4^X) = E(5X^{10}) + E(4^X) = 5p + p(4^1) + (1-p)(4^...
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Random variable with Bernoulli variable

Say $X$ is a variable with $P(X=1)=p$ and $P(X=0)=1-p$. Find $E(Y)$ and $\text{var}(Y)$ for each of the following: (a) $Y=2X$ (b) $Y=X^2$ (c) $Y=2^X$ Okay, so I'm thinking that: (a) $E(Y) = 2p$ ...
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Bernoulli Trial Question in Casella and Berger

Question 3.3 in Casella and Berger states that: Traffic flow can be modeled as a Bernoulli trial (given certain assumptions) Assume that probability of car passing is (fixed) p in any given second ...
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Confidence/credibility intervals for a bernoulli trial

Say we have $$ X \sim \text{Bernoulli}(p). $$ I am interested in finding a 95% confidence and credibility interval. For the credibility, I am assuming a uniform prior, giving me a posterior ...
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Maximum likelihood estimator for distribution with bound constraints

In class, we talked about finding Maximum Likelihood Estimators but there is something I don't think we talked about. How is the MLE different for distributions with constrained bounds? For example, ...
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Geometric distribution with random, varying success probability

I'm really sorry if this question is too basic, but I've been looking for a while and haven't been able to find a convincing response. My statistics background is rather poor. Geometric distribution ...
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Inequality on binomial distribution

Let $X$ denote number of success in $n$ independent Bernoulli trials with probability of success $p$ in each trial. Show that, $$\mathbb{P}[X\ge r] \le \frac{r(1-p)}{(r-np)^2}, \quad if \quad r>np$$...
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simulating dependent bernoulli variates

I have two binary random variables, say $y_A\sim Bernoulli\left(p_A\right)$ and $y_B\sim Bernoulli\left(p_B\right)$ that represent probability of default. I need to superimpose a sort of dependency on ...
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Detection of unmodeled dependence

Given 2 joint groups Bernoulli trials: $X_i \sim p_i$ and $Y_i \sim q_i$, where $(X_i, Y_i)$ are binary outcomes of the $i$-th experiment, and $(p_i, q_i)$ are the probabilities of $X_i=1$ and $Y_i=1$ ...
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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 -\...
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How to compute variance of squared binomial RV?

If $T$ is distributed from a Binomial $\mathcal{B}(n,p)$ distribution, is there a simple way to compute the variance of $$ \frac{T(n-T)}{n(n-1)}=\frac{\sum(X_i-\overline{X})^2}{n-1} $$ where the $X_i$...
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binomial distribution of the same variable at two different time period

I would like to confirm if I have chosen the appropriate test. I'm working on a two wave data sample. There is 750 observations and the participants are exactly the same (cohort study). I want to ...