Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The Bernoulli distribution is a special case of the binomial distribution in which we have precisely one trial with two possible outcomes (with fixed probabilities of occurring).

Use this tag for questions regarding applications of Bernoulli random variables, or for computing quantities that can be derived from Bernoulli random variables - including expectation, standard deviation, and moments.

148 questions
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What is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure?

The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $\mu$ is essentially defined by a collection of…
fromscratch
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Let $X_i\sim\text{Ber}(\frac{1}{2})$ be i.i.d. and $Y_i=\max(X_i,X_{i+1})$ and $Z_n=\sum_{i=1}^{n}Y_i$. Find $\text{E}(Z_n)$ and $\text{Var}(Z_n)$

Problem: Let $X_1,X_2,\dots$ be independent random variables with $X_i\sim\text{Ber}(\frac{1}{2})$. Let $Y_i=\max(X_i,X_{i+1})$ and $Z_n=\sum_{i=1}^{n}Y_i$. Find $\text{E}(Z_n)$ and $\text{Var}(Z_n)$. My solution: We have…
User
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7
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1 answer

Lower bound for the distribution of the sum of independent Bernoulli variables ( not i.i.d)

Suppose we have $X_1,\cdots,X_n$ be independent Bernoulli random variables with $n$ odd, $P(X_i=1)=p_i$ and $p_i \in [1/2,1]$ for all $i=1,\cdots,n$. I want to prove that: \begin{equation} P(\sum_{i=1}^n X_i\geq\frac{n+1}{2}) \geq…
user574579
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5
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How to interpret this limit of probabilities from different sample spaces being $\frac{1}{e}$?

A student asked me this probability question today. Suppose that a Bernoulli trial with probability of success $\frac{1}{n}$ is repeated $n$ times. What is the probability of no successes? A simple counting argument shows the answer is…
beanstalk
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5
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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
user1231900
5
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1 answer

Prove formally that $\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$ converges to $N(0,1)$

I'm not able to prove that $$\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\!\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}}$$ converges in distribution to $N(0,1)$. Here, $\hat{p}_1$ and $\hat{p}_2$ are independent estimations of the same…
moqui
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4
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Expected rank of a random binary matrix with Bernoulli probability p?

Let $M \in \mathbb{R}^{m,n}$. The entries are in {$0, 1$} (with the value $1$ having probability $p$, and the value $0$ having probability $(1-p)$). What is the expected rank of $M$? Follow-up: how does this change if the entries are in {$-1, 1$}…
user3667125
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4
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1 answer

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 $\left(\frac{\lambda}{n}\right),\left\{X_i\right\}$ and $\left\{Y_i\right\}$ are…
Debu
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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 $\dfrac{\sum_{k=1}^{n}X_{k}}{\sum_{k=1}^{n}p_{k}}$ converge to $1$ in probability? Well, the first…
Dovahkiin
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4
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1 answer

Anti concentration inequality for sum of dependent Bernoulli random variables

Thanks in advance for the help. I have $n$ dependent and identically distributed Bernoulli r.v.s $X_1, \dots, X_n$ with success probability $p$. Consider $X = \sum_{i=1}^n X_i$. I know its first ($\mu = \mathbb{E}[X]$) and second ($\nu =…
CuriousGuy
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4
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1 answer

An upper bound of $\mathbb{P}(|S_n - \log n| \geq C \log n)$, where $S_n$ is a sum of $n$ independent Bernoulli$-\frac{1}{i}$ random variables

Let $(X_i)_{i=1}^n$ be independent Bernoulli random variables with parameter $\frac{1}{i}$. Let $S_n = \sum_{i=1}^nX_i$ and $C > 0$. I need a bound of $$\mathbb{P}(|S_n - \log n| \geq C \log n)$$ and apparently I can use Chebyshev's inequality to…
Vicky
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4
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Sum of non i.i.d. Bernoulli and Le Cam's theorem

Consider a (deterministic) sequence $(p_i)_i$ such that $p_i \in (0, 1)$ and $\sum_{I}^n p_i = T_n \rightarrow T < +\infty$. Then define a sequence of independent Bernoulli random variables $B_i \sim \text{Be}(p_i)$. From le Cam's theorem, I know…
mariob6
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4
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1 answer

A construction of an uncountable product of independent Bernoulli variables

I have an intuitive stochastic process as follows, but not sure how to construct it rigorously on some probability space. Consider the unit interval $[0,1]$, each point $x\in [0,1]$ is associated with an independent Bernoulli($0.5$) random variable.…
shong
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3
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2 answers

Sum of a subset of i.i.d. Bernoulli r.v.s conditional on their total sum

Suppose we have i.i.d. $\xi_1,\dots,\xi_n\sim Bernoulli(p)$ and define $\tau=\sum_{i=1}^n\xi_i,\eta=\sum_{j=1}^{\tau}\xi_j$. Find $\mathbb{E}\eta$. Unlike in some of the similar problems solved here, this one involves dependent r.v.s. I start by…
Meh Mech
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3
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0 answers

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 $\exp(c^2t^2)$ for all $t\in\mathbb{R}$, we would be talking about…
Aryeh
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