document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192049>Question 1192049</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\r<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 #849040 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=849040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's break down the problem step-by-step.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Joint probability density function: f(x, y) = e^(-x-y) for x > 0 and y > 0 \r<BR>\n" );
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document.write( "**a) Find the marginal density of X:**\r<BR>\n" );
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document.write( "* To find the marginal density of X, we integrate the joint density function with respect to y over its entire range:\r<BR>\n" );
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document.write( "   f_X(x) = ∫[from 0 to ∞] f(x, y) dy <BR>\n" );
document.write( "          = ∫[from 0 to ∞] e^(-x-y) dy<BR>\n" );
document.write( "          = e^(-x) ∫[from 0 to ∞] e^(-y) dy<BR>\n" );
document.write( "          = e^(-x) * [-e^(-y)] [from 0 to ∞]<BR>\n" );
document.write( "          = e^(-x) * (0 - (-1)) <BR>\n" );
document.write( "          = e^(-x) \r<BR>\n" );
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document.write( "   Therefore, the marginal density of X is:<BR>\n" );
document.write( "   f_X(x) = e^(-x) for x > 0 \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 is defined as:\r<BR>\n" );
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document.write( "   f_Y|X(y|x) = f(x, y) / f_X(x)\r<BR>\n" );
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document.write( "* Substituting the given values:\r<BR>\n" );
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document.write( "   f_Y|X(y|x) = e^(-x-y) / e^(-x) <BR>\n" );
document.write( "               = e^(-y) \r<BR>\n" );
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document.write( "   Therefore, the conditional density of Y given X = x is:<BR>\n" );
document.write( "   f_Y|X(y|x) = e^(-y) for y > 0\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, E[XY], is given by:\r<BR>\n" );
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document.write( "   E[XY] = ∫∫[over the support of (X, Y)] x * y * f(x, y) dx dy\r<BR>\n" );
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document.write( "* In this case, the support of (X, Y) is x > 0 and y > 0:\r<BR>\n" );
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document.write( "   E[XY] = ∫[from 0 to ∞] ∫[from 0 to ∞] x * y * e^(-x-y) dy dx\r<BR>\n" );
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document.write( "* We can solve this double integral using integration by parts. However, a quicker approach is to recognize that the joint density function is the product of the marginal densities of X and Y:\r<BR>\n" );
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document.write( "   f(x, y) = e^(-x-y) = e^(-x) * e^(-y) = f_X(x) * f_Y(y)\r<BR>\n" );
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document.write( "* This indicates that X and Y are independent random variables. \r<BR>\n" );
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document.write( "* For independent random variables, E[XY] = E[X] * E[Y]\r<BR>\n" );
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document.write( "* We know that X and Y both follow an exponential distribution with parameter 1. The expected value of an exponential distribution with parameter λ is 1/λ.\r<BR>\n" );
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document.write( "* Therefore, <BR>\n" );
document.write( "   E[X] = 1/1 = 1 <BR>\n" );
document.write( "   E[Y] = 1/1 = 1\r<BR>\n" );
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document.write( "* Hence, the first joint moment of X and Y is:<BR>\n" );
document.write( "   E[XY] = E[X] * E[Y] = 1 * 1 = 1\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* Marginal density of X: f_X(x) = e^(-x) for x > 0<BR>\n" );
document.write( "* Conditional density of Y given X = x: f_Y|X(y|x) = e^(-y) for y > 0<BR>\n" );
document.write( "* First joint moment of X and Y: E[XY] = 1\r<BR>\n" );
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document.write( "I hope this comprehensive explanation is helpful! </SPAN>\n" );
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