Data Errors

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    The activity for a radioactive decay as a function of time is given by

    A=A_0e^{-\lambda t}

    At time t = 0, we measure A_0, and we also have an estimate of \lambda. At a later time t )assumed known without error) we estimate A from the formula.


    The question:

    1) Construct a formula for the error in A

    (I've done this)

    2)Give approximations for t small and t large. When does the transition between the two regimes occur? Interpret your results

    I don't know how to do the second part. Any ideas greatly appreciated.
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    Nobody?
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    (Original post by Jagwar Ma)
    Nobody?
    I'm not familiar with the terms "t large" and "t small". Do you have definitions for them?

    It's possible someone may be able to assist knowing that extra information, even if I can't.
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    (Original post by ghostwalker)
    I'm not familiar with the terms "t large" and "t small". Do you have definitions for them?

    It's possible someone may be able to assist knowing that extra information, even if I can't.
    No idea. t small could be as t tends to -infinity, and t large could be as t tends towards + infinity?
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    (Original post by Jagwar Ma)
    No idea. t small could be as t tends to -infinity, and t large could be as t tends towards + infinity?
    OK, I can find nothing Googling either, so will assume it's not a technical term.

    What's your formula for the error from part 1)?

    I suspect they're looking for approximations for that formula for the different values of t.
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    (Original post by ghostwalker)
    OK, I can find nothing Googling either, so will assume it's not a technical term.

    What's your formula for the error from part 1)?

    I suspect they're looking for approximations for that formula for the different values of t.
    \frac{\sigma A^2}{A} = \frac{\sigma A_0}{A_0} +(-t \sigma \lambda)^2
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    (Original post by Jagwar Ma)
    ...
    Sorry, I can't help. It's a level of error analysis beyond my knowledge.

    :sad:
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    (Original post by Jagwar Ma)
    \frac{\sigma A^2}{A} = \frac{\sigma A_0}{A_0} +(-t \sigma \lambda)^2
    A further thought, as no one else has responded.

    For large t, your quadratic term will dominate, and we have

    \frac{\sigma A^2}{A} \approx ( \sigma \lambda)^2t^2

    And for small t, the quadratic term will tend to 0, as t tends to zero, so

    \frac{\sigma A^2}{A} \approx \frac{\sigma A_0}{A_0}

    Edit: Anyone more knowledgeable feel free to chip in (rep or PRSOM available ).
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    I agree.

    To begin with ("t small"), you errors will be dominated by the constant term. But as time goes on ("t large") they will be dominated by the quadratic term.

    For a transition point, you might reasonably choose the point where the two terms are equal (i.e. solve for t). Then look at whether your results were made before or after that, and hence what type of error is the dominant one.
 
 
 
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