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Moment Generating Function  Integration Help 
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#1
Dec612, 09:39 AM

P: 4

I am working on a probabilty theory problem:
Let (X,Y) be distributed with joint density f(x,y)=(1/4)(1+xy(x^2y^2)) if abs(x)≤1, abs(y)≤1; 0 otherwise Find the MGF of (X,Y). Are X,Y independent? If not, find covariance. I have set up the integral to find the mgf ∫∫e^(sx+ty)f(x,y)dx dy with both integrals from 1 to 1. I am having trouble integrating this though in order to move on with the problem. I began to try integration by parts and I do not think that is the best route but have no other ideas. If anyone can help, I would greatly appreciate it! 


#2
Dec612, 09:47 AM

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[tex] f(x,y) = \frac{1}{4}(1 + xy(x^2  y^2)), [/tex] or do you mean [tex] f(x,y) = \frac{1}{4(1 + xy(x^2  y^2))}?[/tex] If you mean the first one, just write (1/4)(...) to make it clear; if you mean the second one, write 1/(4(...)). 


#3
Dec612, 09:53 AM

P: 4

Yes, I mean the first one! Thank you.



#4
Dec612, 09:58 AM

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Moment Generating Function  Integration Help
Since you have a 2variable integration, you can integrate first over y (for any fixed x), then integrate the result over x; or you can do it in the other order. Is that what you have tried to do? You really need to supply more details, since we have no way of helping if we do not know where you are stuck. 


#5
Dec612, 10:38 AM

P: 4

I set up the integral
[itex]\frac{1}{4}[/itex][itex]\int[/itex][itex]\int[/itex]e[itex]^{sx+ty}[/itex](1+xy(x[itex]^{2}[/itex]+y[itex]^{2}[/itex]))dx dy Both integrals are from 1 to 1 (I don't know how to show that) I tried distributing xy and using integration by parts with u = 1+(x^3)y +x(y^3) and dv= e^(sx+ty) but not sure that this is the right idea because it would take numerous integrations before reducing to one function and then I would have to do it again for the second integral. Other than that, I haven't tried anything. 


#6
Dec612, 11:19 AM

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[tex] \frac14 \int_{1}^1 e^{sx}\left(\int_{1}^1 e^{ty}(1 + xy(x^2  y^2))\,\mathrm{d}y\right)\,\mathrm{d}x [/tex] Start by working out [itex]\int_{1}^1 xe^{kx}dx[/itex], [itex]\int_{1}^1 x^2e^{kx}dx[/itex], and [itex]\int_{1}^1 x^3e^{kx}dx[/itex]. You're going to need these repeatedly, so you may as well do them once and then write in the values as needed. You may also want to note that [itex]e^k  e^{k} = 2\sinh k[/itex] and that [itex]e^k + e^{k} = 2\cosh k[/itex] 


#7
Dec612, 01:40 PM

P: 4

Ok, thanks!



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