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Math Help - Expected value and variance

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    Expected value and variance

    An electronic system has two components X and Y, and works if just one or both components are functioning. Their lifespan function is written as:

    f(x,y) = 2^e−(x+2y), x > 0, y > 0, otherwise 0

    I've calculted the marginal distributions to be:

    g(x) = e^-x
    h(y) = 2e^-2y

    Next I'm supposed to figure out the expected lifespan for X and Y, plus the variance in lifespan for each. Would appreciate any help!
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    Quote Originally Posted by gralla55 View Post
    An electronic system has two components X and Y, and works if just one or both components are functioning. Their lifespan function is written as:

    f(x,y) = 2^e−(x+2y), x > 0, y > 0, otherwise 0

    I've calculted the marginal distributions to be:

    g(x) = e^-x
    h(y) = 2e^-2y

    Next I'm supposed to figure out the expected lifespan for X and Y, plus the variance in lifespan for each. Would appreciate any help!
    I haven't checked your answers (I assume you can integrate) but now that you have the pdf's for each random variable you just use them in the usual way.
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    Quote Originally Posted by gralla55 View Post
    An electronic system has two components X and Y, and works if just one or both components are functioning. Their lifespan function is written as:

    f(x,y) = 2^e−(x+2y), x > 0, y > 0, otherwise 0

    I've calculted the marginal distributions to be:

    g(x) = e^-x
    h(y) = 2e^-2y

    Next I'm supposed to figure out the expected lifespan for X and Y, plus the variance in lifespan for each. Would appreciate any help!
    Assuming your marginal distributions are correct.

    E(x) = \displaystyle\int^\infty_0 xe^{-x}\,dx = 1
    and E(x^2) = \displaystyle\int^\infty_0 x^2e^{-x}\,dx = 2
    So, Var(x) = 2 - 1^2 = 1

    Same procedure for y.
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    MHF Contributor matheagle's Avatar
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    BY inspection these are two independent exponential rvs
    The paramaters are 1 and 2.
    Without integrating you should know the mean and variance of these two.
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