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Homework Help: Finding the density.

  1. Mar 4, 2014 #1
    These are 2 problems that I can solve, but I don't know how to tell the difference between these 2 when it comes to finding the densities. I just need to know how to tell whichmethod to use to find the density. I do not need the solution.
    1. The problem statement, all variables and given/known data

    Suppose that (X; Y ) is uniformly chosen from the set given by 0 < X < 3 and x < y < root(3x). Find the marginal density fy (y) of Y.

    If X is uniformly distributed on [0, 2], and given that X = x, Y is uniformly distributed on [x 2x], what is P[Y <2]?

    2. The attempt at a solution
    For A), to find the joint density, I integrate 1 dy dx of the shape to get the area. the joint density is then 1/area. This makes sense to me.
    For B), integrating 1 dydx doesn't seem to work and instead, it is just simply combining f(x) and f(y). 1/2 * 1/x to get the density.

    I understand the uniform shortcuts so I know where 1/2 and 1/x came from, but how do I know when to use which method? I,e. how do I know that I need to integrate 1, rather then just multiply f(x)*f(y).
    both tell us that x, y are uniformly distributed; I understand that the difference is that one involves a conditional distribution, so is that the determining factor?

    3. Relevant equations
    if f(x) is uniform on (a,b), the area is b-a. the density would then be 1/(b-a).
    f(x)*f(y) = f(x,y) if independent.
  2. jcsd
  3. Mar 4, 2014 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Neither of your X and Y are independent, so that last equation is useless.

    Think of what density really means:
    [tex] P(x < X < x + \Delta x, y < Y < y + \Delta y) \doteq f(x,y) \Delta x \, \Delta y [/tex]
    (neglecting smaller-order terms like ##(\Delta x)^2,## etc).
    This implies
    [tex] P(y < Y < y+\Delta y | X=x)
    \equiv \lim_{\Delta x \to 0} P(y < Y < y+\Delta y|x < X < x + \Delta x),[/tex]
    and this last conditional probability is
    [tex]P(y < Y < y+\Delta y|x < X < x + \Delta x) =
    \frac{P(x < X < x + \Delta x, y < Y < y + \Delta y)}{P(x <X < x + \Delta x)}[/tex]
    In other words, for small ##\Delta x, \, \Delta y## we have
    [tex] P(x < X < x + \Delta x, y < Y < y + \Delta y) \doteq
    P(y < Y < y+\Delta y | X=x) \cdot P(x <X < x + \Delta x)[/tex]
    You can take it from here.
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