Marginal densities of a Probability

[RET80]

Homework Statement

Y₁ and Y₂ have a joint probability density function given by:
f(y₁,y₂) = {8y₁y₂², 0<=y₁<=1, 0<=y₂<=1, y₁²<=y₂
0, Elsewhere

Homework Equations

f₁(y₁) =ʃ f(y₁,y₂) dy₂
f₂(y₂) =ʃ f(y₁,y₂) dy₁

For E(Y) (later, discussed in part 3):
E(Y₁) = ʃ y₁f(y₁,y₂) dy₁
E(Y₂) = ʃ y₁f(y₁,y₂) dy₂

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

The Attempt at a Solution

I attempted both marginal functions and set the limits of integration as follows:
for f₁(y₁), limits of integration were: y₁ to 1
for f₂(y₂), limits of integration were: sqrt(y₂) to 0

With those limits of integration set up, I then solved for both separately and received the answers:
f₁(y₁) = 8/3y₁ - 8/3y₁⁷
f₂(y₂) = 4y₂³

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(Y₁) and E(Y₂) 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.

 

[LCKurtz]

Homework Statement

Y₁ and Y₂ have a joint probability density function given by:
f(y₁,y₂) = {8y₁y₂², 0<=y₁<=1, 0<=y₂<=1, y₁²<=y₂
0, Elsewhere

Homework Equations

f₁(y₁) =ʃ f(y₁,y₂) dy₂
f₂(y₂) =ʃ f(y₁,y₂) dy₁

For E(Y) (later, discussed in part 3):
E(Y₁) = ʃ y₁f(y₁,y₂) dy₁
E(Y₂) = ʃ y₁f(y₁,y₂) dy₂

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

The Attempt at a Solution

I attempted both marginal functions and set the limits of integration as follows:
for f₁(y₁), limits of integration were: y₁ to 1
for f₂(y₂), limits of integration were: sqrt(y₂) to 0

With those limits of integration set up, I then solved for both separately and received the answers:
f₁(y₁) = 8/3y₁ - 8/3y₁⁷
f₂(y₂) = 4y₂³

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(Y₁) and E(Y₂) 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.

You have them both correct. But remember, they are both valid for their variable from 0 to 1, and 0 elsewhere. So when you calculate expected values your integrals should go from 0 to 1; no variables in the limits or answers.