Question

In a city, the daily number of traffic accidents follows a Poisson distribution with \[\lambda=9\]. What is the probability that the total number of traffic accidents exceeds 1000 in 100 days?

Answer 1

@Zarkon can we change to \(\lambda = 900\)?

Answer 2

One of my solutions is to have:
\[\lambda=9*100\]
Because the interval considered in this case is 100 days. Then I calculate for: \[P(X>1000)\]. But the problem with having Lambda as 900 is that it is too huge. P(X>1000)=1.

Answer 3

how do you calculate \(P(x\geq 1000)\)? using poisson? usually "greater than ..." means calculate everything up to it and subtract from one. must be another formula for this. i will try to look it up

Answer 4

Precisely. Here's what I have done:
\[P(X>1000)=1-\sum _{ i=0 }^{ 1000 }{ poisson(i,\quad \lambda =900) } \]

However, I can't calculate possion up to 1000. It is too huge. Not even my calculator can do it. I used an online calculator to do it but the problem is I believe this question shouldn't require me to do such calculations that require a powerful calculator like a computer.

Answer 5

And the value returned from Mathematica for calculating the sum from 0 to 1000 of the poisson function is 0.9995093672673, which is almost 1. So, I think my answer is still wrong.

Answer 6

yeah i think there is another way of computing \(P(x\geq k)\) for poisson using variance, but it might require some table
i am trying to look it up in a third book

Answer 7

ohh thanks! I didn't know there is another way to compute poisson using variance. But if we need a table, would a table list up to 900? The variance for poisson is the same as its expectation, which is Lambda.

Answer 8

yeah each of these damned books has poisson all over the place
still looking.

Answer 9

haha.... okay sure... thanks! i'm still trying too.

Answer 10

use a normal approximation

Answer 11

i have an idea, probably a bad one
you expect 900 accidents, and want to know what is the probability you get over 1000
reduce all by a factor of 100 and change to you expect 9 accidents, what is the probability you get more than 10
this seems to be done by tables, one of which i found on line, where it will give you this

Answer 12

@Zarkon so we need a table for that too right?

Answer 13

My calculator can do it

Answer 14

I have tried this idea too. The probability is around 0.7 something. There is a formula for poisson.

Answer 15

But with this idea, the interval considered for the probability computed would be for each day, not 100 days any more?

Answer 16

i found this
http://www.wku.edu/~david.neal/statistics/discrete/poisson.html
which tells you how to do it with a calculator, but if my idea is right, then i found a table that says
for \(\lambda =9\) \(P(x<10)=0.70599\)

Answer 17

if it is valid to reduce all by 100, and it seems like it might be, then your answer would be
\(1-.70599\) but again this is relying on a table

Answer 18

sorry but every other example i can find sums up all the others and subtracts from one, as you did at first

Answer 19

yea, it seems pretty valid. thanks! :)

My only concern is if this would be computing for an interval of 1 day instead of 100 days? Because usually, I remember I would adjust the value of Lambda according to the required interval specified in the problem. But in this case, adjusting the Lambda seems too huge. Adjusting the interval value gives a more believable value but casts doubts on the interval that probability value computed for.

Answer 20

*casts doubts on the interval that the probability value is computed for.

Answer 21

i wish i knew. my only answer is that this would be an approximation, and perhaps a bad one.
sorry i cannot be more help, stumped on this one, but i would go with reducing and computing the approximation

Answer 22

do you know what the answer is supposed to be?

Answer 23

Thanks a lot, @satellite73. :)

Answer 24

yw, sorry it was not better

Answer 25

@Zarkon, unfortunately, I don't know the correct answer for this question. I am not given. :(

Answer 26

what don't you like about the answer Mathematica gave you?

Answer 27

Because I won't have Mathematica in the exam. :(
This question suppose to be an exam question.

Answer 28

will you have tables?

Answer 29

will you have a normal table?

Answer 30

yea, I will have normal tables. Can I approximate poisson with normal?

Answer 31

I'm only aware that I could approximate binomial with normal and poisson. But I don't know if I could approx for poisson with normal.

Answer 32

yes...quite accurately with continuity corrections

Answer 33

\[P(X>1000)=P(X>1000.5)\]

\[=P\left(\frac{X-900}{\sqrt{900}}>\frac{1000.5-900}{\sqrt{900}}\right)\]

\[=P\left(Z>\frac{1000.5-900}{\sqrt{900}}\right)\]

Answer 34

But with this, I have to make an assumption that the original data follows a normal distribution? Otherwise, how can I be sure that the data is close to a normal distribution?

Answer 35

No...

here you essentially have the sum of 100 iid random variables. By the CLT it will be approx normal

Answer 36

that is how you get \(\lambda=9\cdot 100\)


if \(X_1,X_2,\cdots X_n\) are poisson(\(\lambda\)) then

\[\sum_{k=1}^{n}X_k\sim\text{poisson}(\lambda\cdot n)\]

Answer 37

ohhh.....yeah!! This sounds very logical. And it looks like a very workable hand method.!

Answer 38

good

Answer 39

sorry i didn't jump into the conversation earlier...I'm grading now...man I hate to grade.

Answer 40

It is through CLT, that this whole poisson becomes close to normal. And therefore, the approximation can take place. It is not exact but close because of the CLT, n more than 30. Thank you so much for your help! :D

Answer 41

haha....thanks! it's okay. I am very happy to learn something from you.
And, grading? You mean like grading students' assignment?

Answer 42

Complex analysis exams

Answer 43

@satellite73 Zarkon gave a pretty good explanation. Just thought you may be interested to learn about it too.

Answer 44

oh wow... Sounds like a difficult subject. haha... thanks for your help!

Answer 45

grading is the worst. guess what i am doing?

Answer 46

it is the worst.....worst part of the job..

I hope you are not grading :)