document.write( "Question 1086742: Suppose the joint probability density function of (X, Y) is given by
\n" ); document.write( "F(x,y)=(2x+y)/210, 2≤x≤6;0≤y≤5
\n" ); document.write( " = 0, otherwise\r
\n" ); document.write( "\n" ); document.write( "Show that
\n" ); document.write( "∫_(-∞)^∞▒∫_(-∞)^∞▒〖f(x,y)dxdy=1〗 \n" ); document.write( "

Algebra.Com's Answer #701032 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Since the function equals zero outside the rectangle, we can reduce the limits of integration to,
\n" ); document.write( " $\"2%3C=x%3C=6\"$
\n" ); document.write( " $\"0%3C=y%3C=5\"$
\n" ); document.write( "

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

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