Binominal distr with a twist !

Hi! I'm a newbe to this forum, hope you can help!

I'm looking at probability function for the sum of flushes during a fixed hour at night (this would be the sum for a district).

And let's say I measure in 10 min interval, that would make 6 values.

Let's say that I know there are 10 people up during this hour, and each could flush in all of those timeslots then:

Y1=X1+X2..+X10

Y2=X1+X2..+X10

...

Y6=X1+X2..+X10

Y1 is the sum of flushes for timeslot1,

all X are equal distibuted with p=1/6, and independent

Then Y1 would be Bin(n,p)=Bin(10;1/6) right?

E[Y1]=np = 10*1/6 = 1,6666.. Var[Y1] =np(1-p) = 1,3888..

If I'm looking att a 1hour series,

All Y could have value 0 if no one flushes, and 10 for that matter.

One example of outcome of the first two timeslots

Y1=1 + 1 +...+1

Y2=1 + 1 +...+1

__Now to the twist:__

If each flushes once and an only once , how to I go about that?

Y1=1+ 0+.. +0

Y2=0+ 0+..+1 (if X1 has been 1 the other must be 0)

All X are independent from each other in the same timeslot (different persons)

But all Y are now dependent since you can and must only flush once

Not knowing what to do I've been fooling around with Hypergeometric Hyp(N,n,m)

and sum of Hypergeometric, saying flush is 1 white ball and the other 5 black.

But I'm on very thin Ice here, I don't know what to do.

How do I get Y2? How do I get Var[Y1]?

E[Y1] would still be 1/6 i guess.

Hope you can help, I read some stocastics a long long (long) time ago..

Re: Binominal distr with a twist !

Just to confirm, before deciding what distribution fits your situation, when you say they flush "once and only once" is it not possible for someone to flush 0 times in an hour?

Re: Binominal distr with a twist !

Thanks for your reply!

No, flushing is just once and zero in all other timeslots for each Xn

One realisation could be (With Y=x1+x2+x3+x4 to make it shorter)

1 =1 0 0 0

1 =0 1 0 0

0 =0 0 0 0

2 =0 0 1 1

0 =0 0 0 0

0 =0 0 0 0

E[y1+y2+..+y6] would be completely deterministic, 4 every time

and E[y1] would be 4/6 (non deterministic)

In my case with X1+X2+..+X10 same principle 10 and 10/6

Re: Binominal distr with a twist !

Re: Binominal distr with a twist !

Suppose you are trying to find Y_{k}. And suppose j people have not flushed before the k^{th} interval.

There are $\displaystyle 10Cj$ combinations of j people from 10. 10Cj is the Combination function

j people must have not flushed k-1 times. The probability of that is $\displaystyle (\frac{5}{6})^{j(k-1)}$

10-j people must have flushed already.

$\displaystyle \text{Probability of having flushed + Probability of not having flushed = 1}$

For one person

$\displaystyle \text{Probability of having flushed} + (\frac{5}{6})^{k-1}=1$

$\displaystyle \text{Probability of having flushed} = 1-(\frac{5}{6})^{k-1}$

Then the probability that 10-j people have not flushed is

$\displaystyle \Big(1-(\frac{5}{6})^{k-1}\Big)^{10-j}$

Therefore the probability that j out of 10 people have not flushed k-1 times is $\displaystyle (10Cj)(\frac{5}{6})^{j(k-1)}\Big(1-(\frac{5}{6})^{k-1}\Big)^{10-j}$

Given that j have not flushed, the probability that i people will flush in the k^{th} interval is $\displaystyle (jCi)(\frac{1}{6})^i(\frac{5}{6})^{j-i}$

j can be anywhere between i and 10. So the probability that i people will flush in the k^{th} interval is

$\displaystyle \sum_{j=i}^{10} (jCi)(\frac{1}{6})^i(\frac{5}{6})^{j-i}(10Cj)(\frac{5}{6})^{j(k-1)}\Big(1-(\frac{5}{6})^{k-1}\Big)^{10-j}$

This simplifies a bit to

$\displaystyle \sum_{j=i}^{10} \frac{10!}{(10-j)!} \times \frac{1}{(10-j)!i!} (\frac{5}{6})^{jk-i} \Big(1-(\frac{5}{6})^{jk-i}\Big)^{10-j}(\frac{1}{6})^i$

And that is the expression for the probability that Y_{k} =i

Re: Binominal distr with a twist !

Quote:

Originally Posted by

**Shakarri** $\displaystyle \sum_{j=i}^{10} \frac{10!}{(10-j)!} \times \frac{1}{(10-j)!i!} (\frac{5}{6})^{jk-i} \Big(1-(\frac{5}{6})^{jk-i}\Big)^{10-j}(\frac{1}{6})^i$

I made a mistake but we can't edit posts after a few hours. It should be this

$\displaystyle \sum_{j=i}^{10} \frac{10!}{(10-j)!} \times \frac{1}{(j-i)!i!} (\frac{5}{6})^{jk-i} \Big(1-(\frac{5}{6})^{jk-i}\Big)^{10-j}(\frac{1}{6})^i$

1 Attachment(s)

Re: Binominal distr with a twist !

Hi !

Thanks!

I saw your answer first time today!

What is you conclusion compared to the Bionom distribution?

I was so confused a few days ago that I wrote down every combination by hand

taken 4 people and 3 timeslots (3 x 20min)

that is 3^4 = 81 combinations! Here is the first 6

4 1111 3 1110 ......

0 0000 1 0001

0 0000 0 0000

...............

And I arrived at

0 1 2 3 4 sum

16 32 24 8 1 freq

1 4 6 4 1 PascTri

I also made the numbers in Excel

16,00

32,00

24,00

8,00

1,00

I believe this is nothing but the (n over k) here (4 over k) k=0,1,.. 4

with the Pascal triangle nr above

And I checked with the Binom(4,1/3) and got exactly the same numbers.

I came to realise that I took a long ride to arrive at Binom never the less.

There are one differens though.

In the "each person must flush once and only once" I Have a constant sum = 4 over every 3 trials.

Hence it's more "stable" than the Binom(4,1/3) that can have the result sum over 3 trials =0=0+0+0 or 12=4+4+4

I belive in the long run they generate the same "population" but Binom is taking longer time doin so.

Have you come to the same conclusion?

Thanks agan for your interest Shakarri! (Clapping)

Attachment 29341

Re: Binominal distr with a twist !

I realized what is going on. If you assume that each of the 81 combinations has the same probability then it should be a binomial distribution, but they are not all equally common so the expression is more complicated.

Re: Binominal distr with a twist !

Quote:

Originally Posted by

**Shakarri** I realized what is going on. If you assume that each of the 81 combinations has the same probability then it should be a binomial distribution, but they are not all equally common so the expression is more complicated.

"but they are not all equally common"

That's right! thats why the combination sum=4 has probability 1/81 and the sum 0 has 16/81.

I have listed all possible outfall. Can you explain why I can't use Binom?

Re: Binominal distr with a twist !

When you listed every combination you assumed that every one of the 81 outcomes were of equal probability equal to 1/81. There were 16 times when 0 people flushed and 1 time when 4 people flushed so you thought that the probability that 0 flush is 16/81 and the probability that 4 people flushed was 1/81. But this is not true, the probability of each of the 81 outcomes is not equal 1/81 so the probability that 0 flush is not equal to 16/81. The probability of each outcome depends on whether someone had flushed previously so it is not simply 1/81.

The other thing is that $\displaystyle \sum_{j=i}^{4} \frac{10!}{(4-j)!} \times \frac{1}{(j-i)!i!} (\frac{5}{6})^{jk-i} \Big(1-(\frac{5}{6})^{jk-i}\Big)^{4-j}(\frac{1}{6})^i$ does not simplify to $\displaystyle \frac{4!}{n!(4-n)!}(\frac{1}{3})^n(\frac{2}{3})^{4-n}$