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Math Help - Joint probability distribuction

  1. #1
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    Joint probability distribuction

    I am not quite sure about how to solve the following exercise:
    Given that a machine is turned on at a random time (X) (in hours) in a particular day and turned off at another random time (Y) (on the same day) determine the joint probability mass function f(X,Y).

    Well I am certain of a thing! the the X marginal prob. mass function is a uniform one, as to the Y one I am not quite sure of it since y must always be greater than x.

    Well even if I know the Y and X distribution, can I assume they are Independent? well the range of Y depends on x doesnt it make one dependent on the other?
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  2. #2
    Guy
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    You know the marginal distribution of X and the conditional distribution of Y. Multiply the corresponding pdfs to get the answer.

    Strictly speaking, though, it bothers me that they don't specify that by "at a random time" they mean uniformly distributed.
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  3. #3
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    Wth Is the conditional distribution of y? Never heard of it! How can I have it.? I only obo that if I have te two marginal distributions and the variables are independent one of another qe can multiply them to get the houve distribution
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  4. #4
    Guy
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    X and Y are clearly not independent since the possible values of Y depends on X. The idea seems to be that X ~ uniform(0, 24) and that, given X, Y ~ uniform(X, 24).

    You could try calculating the cdf explicitly and taking the mixed derivative to get the pdf if you don't know about conditional distributions yet. Calculate

    P(X \le x, Y \le y) = P(Y \le y | X \le x) P(X \le x).

    Certainly, you must know about at least conditional probabilities.
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  5. #5
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    thank you for your reply, really helpfull.
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