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Math Help - Joint Density Question

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    Joint Density Question

    Suppose that X is a continuous uniform random variable on the interval [-1,2], Y is a continuous uniform random variable on the interval [1,5] and that X and Y are independent random variables.
    a) Specify completely the joint density function of (X,Y)
    b) Find P(X+Y greater than or equal to 3)
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  2. #2
    MHF Contributor matheagle's Avatar
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    The marginal densities are 1/3 and 1/4 on the two intervals.
    The joint density is the product of the marginals.

    Draw the region, it's a rectangle, draw the line x+y=3 and integrate
    (actually calc is not needed) over the appropriate region.
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    Quote Originally Posted by matheagle View Post
    The marginal densities are 1/3 and 1/4 on the two intervals.
    The joint density is the product of the marginals.

    Draw the region, it's a rectangle, draw the line x+y=3 and integrate
    (actually calc is not needed) over the appropriate region.

    I'm still a little confused on the joint density. Is it over a certain interval or something?

    For b), what region do I integrate over? I need greater than or equal to 3, wouldn't the integral give me less than or equal to 3? Or do I integrate over the certain interval where it will give me what I want?
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  4. #4
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    Quote Originally Posted by Janu42 View Post
    I'm still a little confused on the joint density. Is it over a certain interval or something?

    For b), what region do I integrate over? I need greater than or equal to 3, wouldn't the integral give me less than or equal to 3? Or do I integrate over the certain interval where it will give me what I want?
    a) It should be crystal clear that the support of the joint density is the rectangular region defined by -1 \leq x \leq 2 and 1 \leq y \leq 5.

    b) Did you do this?:
    Quote Originally Posted by matheagle
    Draw the region, it's a rectangle, draw the line x+y=3 and integrate
    (actually calc is not needed) over the appropriate region.
    Calculate the area above the line that is inside the rectangle. Hint: It's a trapzium. Again, this should be very clear (especially given what you should have learned from the replies to all your other threads, not to metion what you're meant to be learning in class too).
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