# RARE EVENTS -- Poisson / Binomial -- Need Help

#### jchen123456789

##### New Member
Any help with the problem below is greatly appreciated. :wave:

I have 100 days of observations of a geyser. In those 100 days, the geyser shot up to the height of 30 feet or higher 10 times. (There does not seem to be any pattern in the way the geyser does it. There is no information on how often the geyser shoots up and how high, only that it went over this height 10 times in 100 days. It is possible that it could happen twice in a day, but it is unlikely due to lack of any pattern.) This is all the information I have. All I know is that the event occurred 10 times in 100 days.
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Based on the sample, I assume that the daily frequency (probability) is 10%, and there is an expectation of it happening 10 times in the next 100 days. Right?

1. What is the probability that the true mean is 15 (instead of the observed 10) or 20?

2. What is the lowest number for the true mean so that I can be 99% or 98% confident I am not underestimating the mean? In other words, what is the upper limit for the true mean at 99% or 98% confidence?

3. I also have data for another geyser where it only happened twice in 100 days. The sample is extremely limited, but I have to come to some conclusions for it too.

Thank you in advance! Last edited:

#### jchen123456789

##### New Member
I think this is the wrong forum, so I posted the same question in the other one (Probabulity vs Stat). No intent to spam. Sorry.

E

#### elnaz

##### Guest
hello
i hope another friends will answer to this question too
i should tell to you about relate of poisson and binomial distribution
when n->+Inf and p->0 you can approxiamate binomial with poisson
when n->+Inf and p->.5 you can approxiamate binomial with normal
you know in poisson distribution , parameter of distribution is it's mean and variance, too,
in your question lambda(parameter of poisson) is 10 and mu is .1
1)xi~bin(100,.1) => sum(xi)~bin(1000,.1)
p(xbar=15 or 20)=p(xbar=15)+p(xbar=20)=
p(sum(xi)=15*10)+p(sum(xi)=20*10)
with approxiamte
xi~p(10) => sum(xi)~p(100)
p(xbar=15 or 20)=p(xbar=15)+p(xbar=20)=
p(sum(xi)=15*10)+p(sum(xi)=20*10)
2)for confidence interval , should use approximation with normal distribution
3) xi~bin(n,.02) if n will be biggish so xi~p(2)