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Math Help - sample paths

  1. #1
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    sample paths

    <H2>Suppose that the lifetime of a light bulb is exponentially distributed and that the expected lifetime is 250 hours.
    Suppose that we switch on the light bulb at midnight and then we check at midnight every day afterwards in order to see if the light bulb is still burning.
    Let Xn=1 if the bulb is still burning and Xn=0 if it has burnt out, where n=1,2,3, is the number of days after we start the light bulb.

    Sketch two sample paths of for n=1 to 10




    </H2>
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  2. #2
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    Quote Originally Posted by winganger View Post
    <H2>Suppose that the lifetime of a light bulb is exponentially distributed and that the expected lifetime is 250 hours.
    Suppose that we switch on the light bulb at midnight and then we check at midnight every day afterwards in order to see if the light bulb is still burning.
    Let Xn=1 if the bulb is still burning and Xn=0 if it has burnt out, where n=1,2,3, is the number of days after we start the light bulb.

    Sketch two sample paths of for n=1 to 10



    </H2>
    There are only two states that the bulb can be in 1 it is still burning and 0 it is burnt out.

    It starts at 1 and remains at 1 untill it burns out when it becomes 0, and then remains at 0.

    So any sample path consists of a number of 1 followed by 0's. The number of 1's could be any number from zero to ten, as could the number of 0's (well 10 minus the number of 1's).

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    There are only two states that the bulb can be in 1 it is still burning and 0 it is burnt out.

    It starts at 1 and remains at 1 untill it burns out when it becomes 0, and then remains at 0.

    So any sample path consists of a number of 1 followed by 0's. The number of 1's could be any number from zero to ten, as could the number of 0's (well 10 minus the number of 1's).

    RonL
    Thank you very much.
    I would like to ask one more question,
    how to find the mean of Xn, for n=0,1,2,...
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  4. #4
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    Quote Originally Posted by winganger View Post
    Thank you very much.
    I would like to ask one more question,
    how to find the mean of Xn, for n=0,1,2,...
    Won't it just be

    1 \cdot \Pr(Y \geq n) + 0 \cdot \Pr(Y < n) = .....,

    where Y is the random variable lifetime of bulb (days) .....
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