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Thread: probability with density functions

  1. #1
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    probability with density functions

    A radioactive material emits alpha particles at a rate described by the density function
    f(t) = .1e^(-.1t)
    Find the probability that a particle is emitted in the first 10 seconds, given...
    a) NO particle is emitted in the first second
    b) NO particle is emitted in the first 5 seconds
    c) a particle is emitted in the first 3 seconds
    d) a particle is emitted in the first 20 seconds

    for C and D, i think i'm supposed to integrate f(t) from 0 to 3 (and 0 to 20), which gives the probability of those events happening. Then to find f(t given the event), would it be f(t) / the probabilities of the above? But then, if a particle is emitted in the first 3 seconds, wouldn't the probability be 1? And I'm not sure how to set up the integral at this point for part D (the 20 seconds), because I'm unsure of the bounds.

    I have no idea where to begin with A and B, a little push in the right direction would be great! Thanks
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  2. #2
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    If it were me, I'd do the five integrations and then think about it.

    $\displaystyle \int_{0}^{10}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.632$

    $\displaystyle \int_{0}^{1}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.095$

    $\displaystyle \int_{0}^{5}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.393$

    $\displaystyle \int_{0}^{3}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.259$

    $\displaystyle \int_{0}^{20}\frac{1}{10}e^{-\frac{t}{10}}\;dt\;=\;0.865$

    Now,

    1) What do you know about Conditional Distributions?

    2) What property of the Exponential Distibution might help us out on this one?
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  3. #3
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    The probability of event A, given event B, is the probability of A intersect B over the probability of B...in terms of the PDF, f(t|a) = f(t)/p(a)?
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  4. #4
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    Okay. Now do it.

    You didn't answer Question #2.
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  5. #5
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    I have no idea as to what the answer to #2 would be.
    I'm pretty sure I figured out parts a and b...
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  6. #6
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    So for a and b, i did 1-integral (from 0 to 1, and 0 to 5) then divided f(t) by these values and integrated from 1 to 10 and 5 to 10, i think this is correct.

    However, when I apply this method to c (except not doing 1-integral, since the integral is the probability of the event itself) I keep getting numbers larger than 1, which I know is incorrect
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