Isotopes/uncertainty in measurement

[yxgao]

Hi,
Can someone help with a problem on the gre?

A student makes 10 one-second measurements of the disintegration of a sample of a long-lived radioactive isotope and obtains the following values:
3,0,2,1,2,4,0,1,2,5.

How long should the student count to establish the rate to an uncertainty of 1 percent?

The answer is 5000 s.

I have calculated the mean and variance of the sample to be 2 and 24, respectively, but I don't know how to proceed. Any help would be appreciated!

Thanks

 

[ehild]

Hi,
Can someone help with a problem on the gre?

A student makes 10 one-second measurements of the disintegration of a sample of a long-lived radioactive isotope and obtains the following values:
3,0,2,1,2,4,0,1,2,5.

How long should the student count to establish the rate to an uncertainty of 1 percent?

The answer is 5000 s.



s

The standard deviation of the count rate of radioactive decay is sqrt(N) if N is the average count rate and it is 2/s in the problem. To get uncertainty less than 0.01, sqrt(N)/N <0.01, that is N>10^4. That means t>10^4/2 = 5000 s.

ehild

 

[yxgao]

Is there a textbook or online reference that gives the background on this?
Thanks!

 

[ehild]

Is there a textbook or online reference that gives the background on this?
Thanks!

Look for Poisson distribution in any textbook on Statistics or Probability Theory.
I've found a good page at

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm

ehild

 

[yxgao]

How did you know that it was a poisson distribution? I don't understand physically what is happening.

 

[ehild]

How did you know that it was a poisson distribution? I don't understand physically what is happening.

Is it GRE on Physics or Maths?

The Poisson distribution is used to model the number of events occurring within a given time interval in a random way. The decomposition of a radioactive atom of a huge number of similar atoms is such random event.

I can explain more a bit later but now I am busy...


ehild

 

[yxgao]

It's physics. I had no idea that the Poisson distribution could be used for that. Thanks for the tip :) I wonder if you can find this information in a particle physics textbook like Griffiths or something. I haven't taken particle physics in detail yet so I will probably get a better understanding of it once I do.
Thanks for your help :)

 

[ehild]

It's physics. I had no idea that the Poisson distribution could be used for that. Thanks for the tip :) I wonder if you can find this information in a particle physics textbook like Griffiths or something. I haven't taken particle physics in detail yet so I will probably get a better understanding of it once I do.
Thanks for your help :)

I do not think there is any physical theory behind, and I haven't Griffith's book. It is taught in the frames of radioactivity, and mainly during laboratory practice. When observing a radioactive atom, it is totally uncertain when it will decompose. We know the half -life of the isotope, and that the lifetime of radioactive isotopes follow exponential distribution.
The number of decomposing atoms in unit time is proportional to the atoms present. The probability that an atom decomposes during the next short time interval dt depends only on the length of this interval, proportional to the length, and does not depend how long has the atom survived already.
When we observe the number of counts, that is the number of decompositions during a certain interval, we perform a Bernoulli experiment. We watch N atoms and find out how many of them decomposes during the observation time. If p is the probability that an atom decomposes then the probability that we get k counts is obtained according to the binomial disrtribution. P(k) = N!/[k! (N-k)!]p^k(1-p)^(N-k)
The binomial distribution transforms into a Poisson distribution with parameter lambda=np when N is high and p is low, and this is the case with a long-lifetime isotope.

All this is usually discussed in books on probability or statistics.

ehild