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**1. Homework Statement**

Y

_{1}and Y

_{2}have a joint probability density function given by:

f(y

_{1},y

_{2}) = {8y

_{1}y

_{2}

^{2}, 0<=y

_{1}<=1, 0<=y

_{2}<=1, y

_{1}

^{2}<=y

_{2}

0, Elsewhere

**2. Homework Equations**

f

_{1}(y

_{1}) =ʃ f(y

_{1},y

_{2}) dy

_{2}

f

_{2}(y

_{2}) =ʃ f(y

_{1},y

_{2}) dy

_{1}

For E(Y) (later, discussed in part 3):

E(Y

_{1}) = ʃ y

_{1}f(y

_{1},y

_{2}) dy

_{1}

E(Y

_{2}) = ʃ y

_{1}f(y

_{1},y

_{2}) dy

_{2}

All integrals are set to -infinity to +infinity, which are then adjusted to the boundaries of the density function

**3. The Attempt at a Solution**

I attempted both marginal functions and set the limits of integration as follows:

for f

_{1}(y

_{1}), limits of integration were: y

_{1}to 1

for f

_{2}(y

_{2}), limits of integration were: sqrt(y

_{2}) to 0

With those limits of integration set up, I then solved for both separately and received the answers:

f

_{1}(y

_{1}) = 8/3y

_{1}- 8/3y

_{1}

^{7}

f

_{2}(y

_{2}) = 4y

_{2}

^{3}

Now my question is, are my limits of integration setup correctly for this kind of question/equation. I don't think they are because after this, I solve for E(Y

_{1}) and E(Y

_{2}) and if I'm not mistaken, those must be whole numbers and not numbers with variables and if I use the same limits of integration that I used here, it will not work (or it will and just look awful). Not too sure here.