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help me about this integral ?

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tomshack

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can u help me about this integral ?how can i change the variables ?

44_1162824503.JPG
 

for part (a),(b)

use x-1=u , y+1=v ... then u will get 2 integrations of the form e^(-x^2)dx which is a well known integration its result is as shown below from mathematica integrator

**broken link removed**

u can learn about erf(x) here:

https://mathworld.wolfram.com/Erf.html



... now let's solve


(a)Pr(X>2,Y<0) = ∫∫fX,Y(x,y)dxdy
x:2 to inf , y:-inf to 0

make the variable substitution and solve

(b)Pr(0<X<2,|Y+1|>2) =∫∫ fX,Y(x,y)dxdy

we set fX,Y(x,y)=fX(x).fY(y) (statistically independent)

we do integration on x from 0 to 2

and integration on y we divide it into sum of 2 integrations : 1st from (-inf) to -3 and 2nd from 1 to (inf)

(c) Pr(Y>X)=∫fy(y)dy .... y: x to (inf)

where fY(y)=∫fX,Y(x,y)dx ... x: (-inf) to (inf)

the final solution should be function in x
 

thank u so much ahmed finally i solved the question by means of u
 

u r most welcomed tom ..... so glad i could help :)
 

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