What is Poisson distribution: Definition and 138 Discussions
In probability theory and statistics, the Poisson distribution (; French pronunciation: [pwasɔ̃]), named after French mathematician Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.
Homework Statement
A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given...
Homework Statement
We can approximate a poisson distribution from the normal. Suppose lambda is a large positive value; let X ~ Poisson(lambda) and let X1...Xn be independant identicly distributed from a Poisson (lambda/n) distribution. Then X and X1+...+Xn have the same distribution. Use the...
probability of event occurring -- poisson distribution?
I am the keeper of records for my local Volunteer Fire Dept. I have now collected data for each of our incident calls from the last 3 years and have made some _very_ basic stabs at interesting statistics which you can see at...
Hey guys, I'm kind of stuck on this question.
In a certain town, crimes occur at a Poisson rate of 2.4 per month (i.e. according to a Poisson process with a rate of 2.4 per month). What is the probability of having exactly 2 months (not necessarily consecutive) with exactly 4 crimes during...
Homework Statement
X is a discrete random variable that has a Poisson Distribution with parameter L. Hence, the discrete mass function is f(x) = L^{x} e^{-L} / x!.
Where L is a real constant, e is the exponential symbol and x! is x factorial.
Without using generating functions, what is...
Homework Statement
On average, each of the 18 hens in my henhouse lays 1 egg every 30 days. If I check the hens once per day and remove any eggs that have been laid, what is the average number, μ, of eggs that I find on my daily visits? What is the most probable (whole) number of eggs that I...
Nr of customers arriving at a shop follow Poisson.
In 15, an average of 4 customers arrive.
a)
A customer has just arrived. Then a minute passed and no one arrived. What is the probability of it takoing at least 5 more min. until another customer arrives?
b)
Consider 40 non-overlapping...
Homework Statement
1. Suppose that the number of telephone calls an operator receives from 9:00 to 9:05 A.M. follows a Poisson distribution with mean 3. Find the probability that the operator will receive:
a. no calls in that interval tomorrow.
b. three or more calls in that interval the...
Homework Statement
LEt X have a Poisson distribution with u=100. Use Chebyshev's inequality to determine a lower bound for P(75<x<125)
Homework Equations
Chebyshev's Inequality.
The Attempt at a Solution
I'm really unsure of how to go about calculating this problem. Any help...
A store opens at 8 in the morning. from 8 until 10 customers arrive at poisson rate 6 per hour. Between 10 and 12 they arrive at a poisson rate of 10 per hour. From 12 to 2, the store closes for lunch, Finally from 2 to 5 the arrival rate drops linearly from 10 per hour at 2 to four per hour at...
(Not sure if I should have posted this in the h/w problem section since it's not really hw...just a problem I've faced recently. But if it should be there, I can move it there. )
There are 5 boxes.
Each box may contain a certain amount of marbles (1, 2, 3 etc.) and some have no marbles at...
I need to fit a Poisson distribution to this set of data (no. of counts of radioactive decay)
The number of counts in a fixed time interval was recorded 500 times.
With the number of counts going from 0 - 9 respectively
39
106
130
100
67
34
15
7
1
1
I understand how to use...
Please help with this thanks :)
1.
(a) Define the Poisson probability distribution with mean μ.
(b) Write down the binomial distribution for x successes in n independent trials each with probability p of success.
(c) On average, 0.15% of the nails manufactured at a factory are known to...
Homework Statement
A source of liquid is known to contain bacteria, with the mean number of bacteria per cubic centimeter equal to 3. Ten 1 c.c. test tubes are filled with liquid. Calculate the probability that all 10 test tubes will show growth, that is contain at least 1 bacterium each. (use...
b]1. Homework Statement [/b]
prove that
\sum( (e^(-u)) (u(^(x)) )/x! (from x=o to n ) = \int ( (e^(-y))(y^n) )dy/n! (from u to infinite )
Homework Equations
The Attempt at a Solution
i know that the left is Poisson distribution formula but how to do with the 'sum' ?
and...
Homework Statement
Let N,X1, X2, ... be independant random bariables where ?N has a poission Distribution with mean 3 while X1, X2... each has a poisson distribution with mean 7
Determine E[N \sum^N_{i=1} X_i]
Homework Equations
The Attempt at a Solution
E[N \sum^N_{i=1} X_i]...
Homework Statement
In a Poisson process with intensity λ, let X1 be the time until the first event and let X2 be the time between the first and the second event. Let Y be the time until the second event, that is, Y = X1 + X2. Find the density function f(y).
2. The attempt at a solution...
Homework Statement
An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?
[b]2. Homework Equations [/]...
Homework Statement
Suppose a typographical errors committed by a typesetter occurs randomly. If that a book of 600 pages contains 600 such errors, calculate the probability by using Poisson's distribution.
i) that a page contains no errors
ii) that a page contains at least three errors...
Homework Statement
The number of customers entering a cafe during tea time is known to be poisson distribution with λ = 5.
on a particular day, given that at least 2 customers have entered the cafe during the tea time. what is the probability that at least 1 more customers will enter the cafe...
[SOLVED] Poisson Distribution
Homework Statement
Let X be the number of people entering the ICU in a hospital. From Historical data, we know the average number of people entering ICU on any given day is 5
a) What is the probability that the number of people entering the ICU on any given...
Homework Statement
It rains on 10.3 days in the town in October on average. Let X denote the number of days in October on which it rains. Assume that rain falling on different days can be treated as independent events. (31 days in October).
Write down an expression for the probability that it...
Homework Statement
Suppose X has a Poisson distribution with parameter lambda. Given a random sample of n observations,
Find the MLE of lambda, and hat lambda.
Find the expected value and variance of hat lambda.
Show that hat lambda is a consistent estimator of lambda.
Homework...
Homework Statement
Suppose that X has a poisson distribution with parameter \lambda . Given a random sample of n observations, find the MLE of \lambda , \hat{\lambda} .
Homework Equations
The MLE can be found by
\Sigma^{n}_{i=1} \frac{e^{- \lambda} \lambda^{x_{i}}}{x_{i}!}
= e^{-...
Homework Statement
Phone calls are received at Diane residence have a Poisson distribution with \lambda =2.
a) If Diane takes a shower for 10 min, what is the probability that the phone rings Once or Twice.
b) How long can she shower if the probability of receiving no calls be at most 0.5...
[SOLVED] A Chebyshev interval with a poisson distribution
Geophysicists determine the age of a zircon by counting the number of uranium fission tracks on a polished surface; the number of these uranium fission tracks on this surface follows a Possion distribution. A particular zircon is of...
Doing Physics at University and I have never done poisson distributions.
How the hell do I do it?
The question is...
Show that the data on the number of cavalry deaths in the Prussian Army in the 19th Century are consistent with the poisson probability distribution. The date were accumulated...
Homework Statement
An independent, identically distributed sample, x = (x1, ... , xn) of size n, is drawn from a Poisson distribution, parameter A. We want to test the null hypothesis H0 : A = A1 against the alternative hypothesis H1 : A = A2 where A1 < A2.
Write down the likelihood ratio...
Ok,
If the mean time between a single random event occurring is 6 months then is the most probably month for the third event to occur the 18th month?
Thanks!
Hi, in a Poisson Distribution test, what happens when the amount of time is doubled?
For example, in 1 month, lamda=np and I can calculate the probability of x events happening in that 1month.
However, if the question is changed to 6 months, what will i have to do? Thanks.
Hey, I want to write a Computer Simulation in C++, which simulates the development of a DNA sequence with a probability to mutate x in one "generation". I do have a variable number (=n) of copies of this DNA. Now one might think, to simulate the mutation by simply:
sum(n*Poisson distributed...
Hello I'm Presented with the following Poisson distribution question
P(X = x) = \frac{e^{-\lambda} \cdot \lambda^{x}}{x!}
where x \in (1,2,3,\ldots) and \lambda > 0
Then I'm suppose to show that the above can be re-written if
P(X \leq 1) = 1 - e^{- \lambda}
Any idears on how I...
Could anybody attempt to solve this probability question? It incorporates the Poisson Distribution. Thank You.
A company finds that it issues a mean of 7 pairs of earplugs a week to any employee. What is the probability that the number of pairs taken by any employee is 9 per week? (Using the...
so flaws in metal produced by high temperatures occur at a rate of 1 per 10 square feet. what is the probability that there is 3 or more flaws in a 8 x 5 feet.
ok, so I know we need to use poisson disstribution on this, e^-np * np^k/k!.
howver, I don't know my np.
so 1 per 10 square...
ok, so on average, there is a chromosome mutation link once every 10,000 baby births.
approximate the probability that exactly 3 of the next 20,000 babies born will have the mutation.
so using poisson distribution, I let
p = 1/10,000
n = 20,000.
and use formula (e^(-np) * (np)^k /...
Hello In my text the following question is posed:
ON a city street, car backfires are heard 8 times per hour. Use the poisson distribution to find an exact expression for the prob. that a car backfire is heard at most once in a given hour. Do not simplify or evaluate your answer.
Now...
If you have a lottery (Megamillions) and you sell 20,000,000 tickets, the probability of them all losing is given by:
(135,145,919/135,145,920)^20,000,000 = 0.862448363
A close approximation is given by:
e^-(20,000,000/135,145,920) = 0.8624413
I just learned this from a book. That's...