Questions tagged [poisson-distribution]

For questions relating to Poisson distributions in probability theory. To be used with [probability] or [probability-distributions] tag.

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Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
Esteban Crespi's user avatar
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2k views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
Landon Carter's user avatar
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129 views

A weighted sum of independent Poisson random variables $X_1 + 2X_2 + 3X_3+\dots+nX_n$

I have that for $1 \leq i \leq n$, the mutually independent random variables $$X_i \sim \text{Poisson}(\mu_i)$$ Then what is the distribution of $$Y \sim \sum_{i=1}^{n}i X_i$$ It looks a bit like an ...
apg's user avatar
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P.G.F. of total progeny in a Poisson branching process

Let $c<1$. Let $X$ be a random variable with distribution: $$\forall k\in\mathbb{N}:\Pr[X=k]=\frac{e^{-ck}\cdot (ck)^{k-1}}{k!}$$ In fact, $X$ is an r.v. describing the total progeny in a Poisson ...
Pois1's user avatar
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When to use Poisson distribution?

I'm still very confused regarding when to use probability distributions. For instance, this is the assumptions to use Poisson distribution, according to Wikipedia: k is the number of times an event ...
Andrew Joplh's user avatar
4 votes
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35 views

Find a compund Poisson variable with characterist function as centered compound Poisson

I know that if $(X_j , j \geq 1)$ is a sequence of i.i.d. process with $\sigma$ being the probability distribution of $X_j's$ and $N \sim \text{Poisson}(\lambda)$ independent of the $X_j's$, then the ...
Fam's user avatar
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Expected value of falling factorials from axioms of Poisson process

Falling factorial, $(x)_n$, is the product of biggest $n$ terms in factorial, $(x)_n = x(x-1)(x-2)\cdot \ldots \cdot (x-n+1)$. Or the number of ways to color the set of $n$ objects into different ...
Roah's user avatar
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4 votes
2 answers
645 views

Distribution of no. of siblings of a random child if the no. of children of a family is Poisson distributed

Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean $\lambda$. Show that the number of ...
qwerty_uiop's user avatar
4 votes
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1k views

Law of large numbers; Poisson distribution

Let $X_n$ be the numbers of job applications at a company in the year $1900+n,n\in\mathbb N$. Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent, identically distributed random variables with ...
Sha Vuklia's user avatar
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MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased? Does there exist an unbiased estimator of $P\{X_{i}=1\...
Zander's user avatar
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162 views

Poisson Process: indepedent increment

Let $\{N(t): t\geq0\}$ be a Poisson process of rate $\lambda$, and let $S_n$ denote the time until the $n_{th}$ event occurs. compute $P(S_3>5|N(2)=1)$ Attempt: Notice that $P(S_3>5)=P(N(5)&...
randy's user avatar
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1 answer
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Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
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What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
stochastic_newbie's user avatar
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sum of dependent bernoulli random variables is poisson?

Suppose you have a sequence of bernoulli variables, $X_n$ with $\frac{1}{n} = \mathbb{P}(X_n = 1) = 1 - \mathbb{P}(X_n = 0)$ and let $S_n = \sum_{k=1}^{n} X_k X_{k+1} $. The goal is that $S = \lim_{n \...
Troubadour's user avatar
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Conjecture about average cluster size in a prime-like sequence

Consider the sequence $u_n=n^a+p_n$, where $a$ is a real constant and $p_n$ is the $n$th prime. Write down the terms in $u_n$ increasing from left to right. From each term, draw a line segment ...
Dan's user avatar
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3 votes
2 answers
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Prove that the following sequences monotonically decrease and increase correspondingly. Since they are bounded, find the limit.

Let $\xi_n$ be a Poisson random variable with $\lambda = n \in \mathbb{N}$. That is $P(\xi_n = k) = \frac{n^ke^{-n}}{k!}$, for $k \in \mathbb{N}_0$. Let $f_+(n) = P(\xi_n \geq n),\ f_-(n) = P(\xi_n &...
Egor's user avatar
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Spherical harmonics - Computing the variance of Poisson noise integrated over $\ell$ on a defined quantity?

It is an astrophysics context but actually, it is mostly a mathematics issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a $C_\...
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331 views

Find the probability that a string of $100$ lights contains at most four defective bulbs using the Poisson distribution

A manufacturer of Christmas tree light bulbs knows that $3\%$ of its bulbs are defective. Find the probability that a string of $100$ lights contains at most four defective bulbs using the Poisson ...
Jessie's user avatar
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3 votes
0 answers
208 views

Reference Request: Good book on parameter estimation for stochastic processes

I have to do some work where I need to estimate the parameters for a poisson process and a Hawkes process from data. I was looking through some of my old probability and stochastic processes textbooks,...
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Closed form for Compound Binomial Poisson PMF.

A compound Poisson process involves events arriving according to a Poisson process with rate $\lambda$. However, with each arrival, instead of there being just one event, we generate a binomial random ...
Rohit Pandey's user avatar
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Approximate independence for fixpoints of random permutations

Let $F_n$ be the random variable that is the number of fixed points of a random permutation on $n$ elements. As $n \to \infty$, the distribution of $F_n$ approaches a Poisson distribution with mean 1....
Geoffrey Irving's user avatar
3 votes
0 answers
139 views

What is $E[\log(Z!)]$ when $Z$ is Poisson with rate $\lambda$?

Suppose $Z$ is a discrete random variable taking values in $\{0,1,2,\ldots\}$, and for $\lambda>0$, $Z$ is distributed as Poisson with rate $\lambda$. It seems a bit cumbersome to evaluate \begin{...
P. N. Karthik's user avatar
3 votes
0 answers
529 views

Time dependent Poisson arrival rate in wait-time theory

A typical Poisson process assumes an average rate $\lambda$ over time which is ignorant to previous events. To calculate how many events occurs after some time period could be done by simply ...
Robin Kramer-ten Have's user avatar
3 votes
0 answers
669 views

poisson process arrivals (insurance company: accidents and time until a claim is reported)

Policy holders of an insurance company have accidents at times of a Poisson process with rate $\lambda$. The distribution of the time $R$ until a claim is reported is random with $P(R ≤ r) = G(r)$ ...
Nikolaos Skout's user avatar
3 votes
0 answers
383 views

Poisson: Conditional Probability on Pizza order

I am not sure about my answer. In particular, part b of the following question. Pizza orders arrive according to a Poisson process of rate 20 per hour. Orders are independently for a vegetarian ...
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Poisson distribution transformation method

Random independent variables $x_1, x_2 \sim \ \ \text{poisson(}\lambda )$ $y=x_1+x_2$ $z=x_1-x_2$ The possible density function $f(y,z)=?$ by using Inverse transformation method. Note that I ...
1190's user avatar
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3 votes
1 answer
1k views

Is there a big chance that Ithaca helps Dryden?

Suppose calls to Dryden Fire Department arrive according to a Poisson Process with rate $0.5$ per hour. Suppose the time $T$ needed to respond to a call, return to the station and be ready for the ...
Landon Carter's user avatar
3 votes
0 answers
976 views

Probability Generating Function of Compound Poisson Process

Let $(N_t)_{t\ge 0}$ be a poisson process with intensity $\alpha > 0$. Let $(X_n)_{n \in \mathbb N}$ be iid real valued random variables that are independent of $N_t$. Let $Y_t = \sum^{N_t}_{k=1}...
WeakLearner's user avatar
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3 votes
0 answers
141 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let $\...
Frank's user avatar
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3 votes
1 answer
937 views

Two independent Poisson processes.

I am trying to prove the result that exactly $k$ occurrences of a Poisson process before the first occurrence of another independent Poisson process is a geometric random variable. \begin{align} &...
Dip's user avatar
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2 votes
0 answers
93 views

$x$ is distributed as Poisson with parameter $\lambda$, prove $\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0$

Let $x\sim\text{Pois}(\lambda)$, prove that for any $\lambda>0$, $$f(\lambda)=\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0.$$ In other words, $$\sum_{x=0}^{\infty}\log(x+0.5)\frac{\lambda^x}{...
zerotrial's user avatar
2 votes
0 answers
74 views

Verifying the inverse Laplace transform for a production-inventory problem: total expected backlogs when demand is Poisson

I am entirely self-taught when it comes to Laplace transforms, and I am seeking an independent opinion on my attempt to work out how to arrive at the below expression (note: I am interested ...
DrEti's user avatar
  • 63
2 votes
2 answers
337 views

Distribution of the time for the first ocurrence in a Poisson Process given the number of events

I am trying to solve this problem: Let N(t) a Poisson Process of rate $\lambda$ where ocurrences are type I with probability $p$. Given that $N(t_0) = n$, what is the distribution of the waiting time ...
José C.'s user avatar
2 votes
0 answers
49 views

Waiting time for Poisson arrival and uniform pickup

At a station, buses leave every $t$ seconds carrying all the waiting people. The arrival times of these people is Poisson distributed with mean $\lambda$. What is the expected waiting time of a ...
muser's user avatar
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2 votes
0 answers
35 views

Proving that $|\mathbb{E}[X^r||^{1/r} = O(r)$, where $r = 1, 2, \dotsc$ and $X \sim$ Poisson($\lambda$).

If $X \sim$ Poisson($\lambda$) and $r = 1, 2, \dotsc$, then I would like to show that $|\mathbb{E}[X^r||^{1/r} = O(r)$. Attempt: For any $r = 1, 2, \dotsc$, $$ \mathbb{E}[X^r] = \sum_{j=1}^r\lambda^j{...
Vicky's user avatar
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2 votes
0 answers
24 views

How to find variance in Poisson?

The number of ice creams sold per hour from Mr Fishy’s van is observed to be a Poisson random variable with parameter 𝜆=8. Each ice cream costs two pounds but Mr Fishy has to pay five pounds per hour ...
Shanaya Thompson's user avatar
2 votes
0 answers
204 views

Relating the loss functions of a geometric and Poisson distribution to their compounding version.

For a random variable $X$, the loss function is defined as $$L(r) = \int_{r}^\infty (x-r)f(x)dx,$$ Or, for a discrete distribution, $$L(r) = \sum_{x=r}^\infty (x-r)f(x).$$ Now, we define the compound ...
Steven01123581321's user avatar
2 votes
0 answers
34 views

Are Poisson distributions with low mean heavy-tailed?

It is very apparent to me how using the normal distribution to estimate the probability of large, Poisson-distributed events may lead to significant underestimates of the probability of these events, ...
deppep's user avatar
  • 78
2 votes
0 answers
549 views

Poisson distribution problem with expected number

The number of typing errors made by a typist has a Poisson distribution with an average of three errors per five pages. If a page contains an error, the typist must retype the whole page. What is the ...
Nhung Huyen's user avatar
2 votes
0 answers
41 views

Confidence interval for Poisson distribution

$X_{1}, X_{2}, ..., X_{n}$ is a random sample from $Poisson(\lambda)$ population. I need to show that when sample size n is large, the approximate two-sided (1-$\alpha$)% C.I. is $$ \left[ \bar{x} + ...
blurrrrrr's user avatar
2 votes
0 answers
62 views

Poisson vs. Exponential distirbution

What is the relationship between Poisson and Exponential distribution ? How can I obtain one from the other ?
user122424's user avatar
  • 3,906
2 votes
0 answers
58 views

Central Limit Theorem - Asymptotic approximation to sums of Poisson random variables and Binomial random variables

Question: Let $X_1, X_2, ...$ be a sequence of iid $\textrm{Poi}(5)$ variables. Let $Y_1, Y_2, ...$ be a sequence of iid $\textrm{Bin}(10,\dfrac{1}{2})$ variables. Define the sums $S_n = \sum_1^n X_i$ ...
Balkys's user avatar
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2 votes
0 answers
55 views

Durek Probability pg. 175 Integration & probability question

I cannot understand why the following expression is true $\operatorname{Pr}(A \in D, B \leq t)=\int_{0}^{t} f_{B}(z) \operatorname{Pr}(A \in D \mid B=z) \mathrm{d} z$ (I) From my limited understanding,...
stateless's user avatar
  • 121
2 votes
0 answers
82 views

Compound Poisson Process and a distribution

I have been asked the following question: Let $\{X_t\}_{t\geq 0}$ a compound Poisson process with $\lambda=2$ where $Y_i\sim \mathrm{Exp}(1)$. We define $\tau=\inf\{t\geq 0 : X_t\geq 10\}$. Find the ...
Miguel Angel Andrade Velázquez's user avatar
2 votes
0 answers
66 views

Unbiased estimator of $1/(1-a)$ when random variables are Poisson(a)

Let $X_1,...,X_n$ be Poisson with parameter $a$. I am looking for a unbiased estimator of $h(a)=\frac{1}{1-a}$ Let $T$ be a statistic and $g(t)$ be it's pmf. Then if we have $E(T)= h(a)$ then $T$ is ...
Infinity_hunter's user avatar
2 votes
0 answers
29 views

How to solve for the value of a discrete random variable without brute force? (Poisson Distribution)

Say I had a question like so: $X$ ~ $Po(3.1)$ Given $P(X < a) = 0.8$ (1 s.f) Solve for $a$ I can brute force it by summing the value of the probability mass function at $X=0$, $X=1$, $X=2$ etc. up ...
Starry Sky's user avatar
2 votes
1 answer
385 views

Poisson Variable is Independent of sum of Bernoulli Variables

Let $(X_n)_{n\geq 1}$ be i.i.d. Bernoulli random variables with parameter $p\in(0,1)$. Let $N$ be a Poisson random variable with parameter $\lambda>0$. Assume $N$ is independent from $(X_n)_{n\...
Joe Shmo's user avatar
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2 votes
0 answers
241 views

Probability of a egg surviving with a Poisson distribution

The probability that a bird deposits $r$ eggs in its nest is given by the Poisson distribution with parameter $\lambda$. Assume that the probability of a egg to survive in nature is $p$ and that the ...
mathiscool's user avatar
2 votes
0 answers
1k views

Poisson distribution for calculation of number of calls

A roadside assistance center receives 2 calls every half hour on average. a) Find the probability for this center to get at least 3 calls in two hours. b) If the center received 3 calls in the first ...
Samuel's user avatar
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2 votes
0 answers
157 views

Relationship between Binomial and Poisson Distributions

Suppose I have an average of 10 cars passing through a street in an hour. 1 Hour = 3600 sec Assuming 1 sec = single trial of a binomial distribution I have a binomial distribution with parameters n = ...
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