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Questions tagged [poisson-distribution]

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

The PMF of a random variable $X$ distributed according to the Poisson distribution with parameter $\lambda > 0$ is the following: $$\Pr\left[X=k\right]=\frac{\lambda^k \exp(-\lambda)}{k!}\;,\; k\geq 0$$ This distribution describes the number of independent events occurring with constant rate in some unit time, the average being $\lambda$ events per unit.

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Help me reconcile my own Poisson distribution calculation to that on a web page.

On a web page here I see the following question: The web page provides this answer: But my own solution is different: Who's right? What's the mistake in the wrong solution? P.S. And by the way, what is the purpose in stating the formula as a…
Chaim
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Throughput, Capacity & Average Waiting Time

thank for your help in advance. I'm dealing with the following problem: Attempt to the solution of the problem: a) If my understanding is right, we have two servers, the first server's throughput rate is $\frac{1}{3} min^{-1}$ and the second…
Ernestas Blaževič
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Poisson Distribution - How does it come to whatever it is right now?

The Question Two cafes are side by side and are open 7 days a week. (a) The first cafe sells cappuccinos at 5 dollars each and has 30 customers a day. Half the customers buy a cappuccino, the rest don't. What is the distribution of dollar sales…
Maisha Kroza
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Bloody Poisson question. $X \sim Po(\lambda). P(X>2) = 0.3$. Find $P(X<2)$.

Let $X$ be a random variable with a Poisson distribution, such that $P(X>2) = 0.3$. Find $P(X<2)$. This is bunched in with some easy Poisson questions in the textbook. I don't see how to answer this easily though. $ e^{-\lambda} +…
Adam Rubinson
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Finding a function of $Pois(\lambda_1)$ and $Pois(\lambda_2)$ such that the distribution depends on $\frac{\lambda_1}{\lambda_2}$ only.

Suppose that $X\sim Pois(\lambda_1)$ and $Y\sim Pois(\lambda_2)$ are two independent Poisson random variables. Can we find a function $f(X,Y)$ of $X$ and $Y$, where $f(X,Y)$ does not involve $\lambda_1$ and $\lambda_2$, such that the distribution of…
ydwang
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Poisson Thinning and Pareto Distribution

was hoping to get some advice on this one. It is a two part question. The number of days(D) in any given month has a Poisson Distrubtion with $\lambda$ =10 days that rain per month. Let $X_1,X_2$... be independent amounts of rain(mm) that are…
csa61302
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Using Poisson to calculate the probability of total number of goals scored in a certain number of matches

I know I can use the Poisson distribution to calculate the probability of x goals being scored in a match, based on the average number of goals scored per game in that particular league. However, how do I calculate the probability of, for example,…
CPM
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Poisson distribution on pizza request

The demand of IT service is a random variable ($X$) having the Poisson distribution $μ=3.4$. The cost of producing each pizza is $\$20$ and each is sold at a selling price of $\$40$. Suppose $5$ pizza are made daily, no pizza is reserved for next…
Simon Leung
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Extra term while proving an expression involving poisson's distribution

In my textbook I have been given a proving question involving Poisson distribution. The question sentence is as follows: If $m$ and $\mu_r$ denote the mean and central $r$th moment of a Poisson distribution, then prove that…
Kartik Watwani
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Conditional Poisson processes with Multiple conditions

Let $\lbrace N(t)\rbrace_{t\geq 0}$ be a Poisson process with intensity $\lambda = 3$. Compute $$P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right].$$ I understand when there is only one condition i.e. $P\left[N(6) = 2 | N(8) = 4\right]$. Since that is…
V.L.
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Poisson Distribution Probability

If the average number is 16, what is the probability that a given trial will trap between 12 and 20 atoms, inclusive? What would this probability be if the distribution is instead Gaussian, with both a mean and a variance of 16? I got the formula…
AbdelXY
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Poisson distribution with joint PMF

The question is regarding a given pair (X,Y) where x = 0,1,2... and y = x,x+1,x+2,... that have joint PMF of Px,y(x,y) = (y ; x)*(e^-1)/((2^y)(y!)) I need to utilize this given equation to get marginal PMF of X and Y, and then for conditional PMF od…
GarageN
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Poisson distribution and price reduction

A store owner has certain existence of an item and he decides to use the following scheme to sell it: The item has a price of \$100. The owner will reduce the price in half for every customer that buys the item on a given day. That way, the first…
paranoidhominid
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if I have a Poisson random variable $X$, how do I find the constant 'k' that makes $P(X=k)$ be max?

well the book suggests to compare with $\mathbb{P}[X=k-1]$, I notice that, when $k$ is equal to the parameter $\lambda$ we have $\mathbb{P}[X=\lambda]={P}[X=\lambda-1]$, $\mathbb{P}[X= \lambda]$ and it's the greatest probability since the…
newtonplusapple
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Sum of geometric and Poisson distribution

Suppose I have $X \sim \mathrm{Geom}(p)$ and $Y=\mathrm{Pois}(\lambda)$. I want to create $Z = X + Y$, where the $X$ begins at $0$ rather than $1$. Is this possible? Then I would calculate the mean and variance.
user1544953
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