document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192415>Question 1192415</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given f(x, y) = e^(-x-y), for x>0 and y>0<BR>\n" );
document.write( "(a)	Find the marginal density of X<BR>\n" );
document.write( "(b)	Find the conditional density of Y given X = x<BR>\n" );
document.write( "(c)	Find the first joint moment of X and Y<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848994 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find the marginal density of X**\r<BR>\n" );
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document.write( "* The marginal density of X, denoted by f<sub>X</sub>(x), is found by integrating the joint density function f(x, y) with respect to y:\r<BR>\n" );
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document.write( "   f<sub>X</sub>(x) = ∫<sub>-∞</sub><sup>∞</sup> f(x, y) dy \r<BR>\n" );
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document.write( "   Since f(x, y) = e<sup>-x-y</sup> for x > 0 and y > 0, and 0 otherwise, we have:\r<BR>\n" );
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document.write( "   f<sub>X</sub>(x) = ∫<sub>0</sub><sup>∞</sup> e<sup>-x-y</sup> dy \r<BR>\n" );
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document.write( "   = e<sup>-x</sup> ∫<sub>0</sub><sup>∞</sup> e<sup>-y</sup> dy \r<BR>\n" );
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document.write( "   = e<sup>-x</sup> [-e<sup>-y</sup>]<sub>0</sub><sup>∞</sup> \r<BR>\n" );
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document.write( "   = e<sup>-x</sup> (0 - (-1)) \r<BR>\n" );
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document.write( "   = e<sup>-x</sup> for x > 0 \r<BR>\n" );
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document.write( "   and 0 otherwise.\r<BR>\n" );
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document.write( "**b) Find the conditional density of Y given X = x**\r<BR>\n" );
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document.write( "* The conditional density of Y given X = x, denoted by f<sub>Y|X</sub>(y|x), is given by:\r<BR>\n" );
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document.write( "   f<sub>Y|X</sub>(y|x) = f(x, y) / f<sub>X</sub>(x) \r<BR>\n" );
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document.write( "   Since we found f<sub>X</sub>(x) = e<sup>-x</sup>, we have:\r<BR>\n" );
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document.write( "   f<sub>Y|X</sub>(y|x) = (e<sup>-x-y</sup>) / (e<sup>-x</sup>) \r<BR>\n" );
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document.write( "   = e<sup>-y</sup> for y > 0 \r<BR>\n" );
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document.write( "   and 0 otherwise.\r<BR>\n" );
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document.write( "**c) Find the first joint moment of X and Y**\r<BR>\n" );
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document.write( "* The first joint moment of X and Y, denoted by E[XY], is given by:\r<BR>\n" );
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document.write( "   E[XY] = ∫<sub>-∞</sub><sup>∞</sup> ∫<sub>-∞</sub><sup>∞</sup> x*y*f(x, y) dx dy\r<BR>\n" );
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document.write( "   = ∫<sub>0</sub><sup>∞</sup> ∫<sub>0</sub><sup>∞</sup> x*y*e<sup>-x-y</sup> dx dy\r<BR>\n" );
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document.write( "   = ∫<sub>0</sub><sup>∞</sup> x*e<sup>-x</sup> [∫<sub>0</sub><sup>∞</sup> y*e<sup>-y</sup> dy] dx\r<BR>\n" );
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document.write( "* Note that ∫<sub>0</sub><sup>∞</sup> y*e<sup>-y</sup> dy is the expected value of an exponential random variable with parameter 1, which is equal to 1.\r<BR>\n" );
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document.write( "   Therefore,\r<BR>\n" );
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document.write( "   E[XY] = ∫<sub>0</sub><sup>∞</sup> x*e<sup>-x</sup> dx\r<BR>\n" );
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document.write( "* This integral also represents the expected value of an exponential random variable with parameter 1, which is equal to 1.\r<BR>\n" );
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document.write( "   **So, the first joint moment of X and Y is E[XY] = 1.**\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The marginal density of X is f<sub>X</sub>(x) = e<sup>-x</sup> for x > 0.<BR>\n" );
document.write( "* The conditional density of Y given X = x is f<sub>Y|X</sub>(y|x) = e<sup>-y</sup> for y > 0.<BR>\n" );
document.write( "* The first joint moment of X and Y is E[XY] = 1.<BR>\n" );
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