## xEnOnn 3 years ago 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?

1. satellite73

@Zarkon can we change to $$\lambda = 900$$?

2. xEnOnn

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.

3. satellite73

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

4. xEnOnn

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.

5. xEnOnn

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.

6. satellite73

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

7. xEnOnn

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.

8. satellite73

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

9. xEnOnn

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

10. Zarkon

use a normal approximation

11. satellite73

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

12. satellite73

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

13. Zarkon

My calculator can do it

14. xEnOnn

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

15. xEnOnn

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

16. satellite73

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 $$\lambda =9$$ $$P(x<10)=0.70599$$

17. satellite73

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

18. satellite73

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

19. xEnOnn

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.

20. xEnOnn

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

21. satellite73

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

22. Zarkon

do you know what the answer is supposed to be?

23. xEnOnn

Thanks a lot, @satellite73. :)

24. satellite73

yw, sorry it was not better

25. xEnOnn

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

26. Zarkon

27. xEnOnn

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

28. Zarkon

will you have tables?

29. Zarkon

will you have a normal table?

30. xEnOnn

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

31. xEnOnn

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.

32. Zarkon

yes...quite accurately with continuity corrections

33. Zarkon

$P(X>1000)=P(X>1000.5)$ $=P\left(\frac{X-900}{\sqrt{900}}>\frac{1000.5-900}{\sqrt{900}}\right)$ $=P\left(Z>\frac{1000.5-900}{\sqrt{900}}\right)$

34. xEnOnn

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?

35. Zarkon

No... here you essentially have the sum of 100 iid random variables. By the CLT it will be approx normal

36. Zarkon

that is how you get $$\lambda=9\cdot 100$$ if $$X_1,X_2,\cdots X_n$$ are poisson($$\lambda$$) then $\sum_{k=1}^{n}X_k\sim\text{poisson}(\lambda\cdot n)$

37. xEnOnn

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

38. Zarkon

good

39. Zarkon

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

40. xEnOnn

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

41. xEnOnn

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

42. Zarkon

Complex analysis exams

43. xEnOnn

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

44. xEnOnn

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

45. satellite73

grading is the worst. guess what i am doing?

46. Zarkon

it is the worst.....worst part of the job.. I hope you are not grading :)