1. ## Density Question

1) Assume that (X,Y) has the following density function
f(x,y) = 2e^-(x+y) when 0 < x < y
a) Find the marginal density function fx(x) and fy(y)
b) Find the probability that P (Y < 2X)
c) Find P(Y < 1 given X < 1)
d) Find P(Y < 1 given X = 1/2). Also the E(Y given X=1/2).

2. a) $\displaystyle f_X(x)=2e^{-x}\int_x^{\infty}e^{-y}dy$ and $\displaystyle f_Y(y)=2e^{-y}\int_0^ye^{-x}dx$

b) Draw and integrate

c) Use $\displaystyle P(A|B)={P(AB)\over P(B)}$

d) Find f(y|x) and integrate.

3. For b), what do I draw?

For c), isnt it P(A union B)/ P(B)? And how do I know what something like P(B) is? Integrate from 0 to 1 with X? Not exactly sure what to ask, but I'm still confused.

And for d), for the expectation part, do I just integrate with y * f(Y given x)?

4. Originally Posted by Janu42
For b), what do I draw? Mr F says: Integrate f(x, y) over the region that is under the line y = 2x but above the line y = 0 (that is, the x-axis).

For c), isnt it P(A union B)/ P(B)? Mr F says: No it's not. Did you look in your class notes at all. It's intersection (which is what matheagle's notation says) not union.

And how do I know what something like P(B) is? Integrate from 0 to 1 with X? Mr F says: Yes. Use the marginal density function.

Not exactly sure what to ask, but I'm still confused.

And for d), for the expectation part, do I just integrate with y * f(Y given x)? Mr F says: What do your class notes say to do?
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5. Originally Posted by mr fantastic
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Sorry, yeah I meant intersection, but OK. I was just confused because it says AB and doesn't have the intersection symbol but I get it thanks.