SOLUTION: Suppose the joint probability density function of (X, Y) is given by F(x,y)=(2x+y)/210, 2≤x≤6;0≤y≤5 = 0, otherwise Show that ∫_(-∞

Algebra ->  Probability-and-statistics -> SOLUTION: Suppose the joint probability density function of (X, Y) is given by F(x,y)=(2x+y)/210, 2≤x≤6;0≤y≤5 = 0, otherwise Show that ∫_(-∞      Log On


   

Question 1086742: Suppose the joint probability density function of (X, Y) is given by
F(x,y)=(2x+y)/210, 2≤x≤6;0≤y≤5
= 0, otherwise
Show that
∫_(-∞)^∞▒∫_(-∞)^∞▒〖f(x,y)dxdy=1〗
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Since the function equals zero outside the rectangle, we can reduce the limits of integration to,
2%3C=x%3C=6
0%3C=y%3C=5

int%28int%28f%28x%29%2Cdx%29%2Cdy%29=%281%2F210%29%2Aint%28%28x%5E2%2Bxy%2BC%29%2Cdy%29
Evaluating,
x%5E2%2Bxy%2BC=%286%5E2%2B6y%2BC%29-%282%5E2%2B2y%2BC%29
x%5E2%2Bxy%2BC=%2836%2B6y%29-%284%2B2y%29
x%5E2%2Bxy%2BC=32%2B4y
So,
int%28int%28f%28x%29%2Cdx%29%2Cdy%29=%281%2F210%29%2Aint%28%2832%2B4y%29%2Cdy%29
int%28int%28f%28x%29%2Cdx%29%2Cdy%29=%281%2F210%29%2A%2832y%2B2y%5E2%2BC%29
Evaluating,
32y%2B2y%5E2%2BC=%2832%285%29%2B2%285%29%5E2%2BC%29-%280%2B0%2BC%29
32y%2B2y%5E2%2BC=160%2B50
32y%2B2y%5E2%2BC=210
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.
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int%28int%28f%28x%29%2Cdx%29%2Cdy%29=%281%2F210%29%2A%28210%29
int%28int%28f%28x%29%2Cdx%29%2Cdy%29=1