document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172515>Question 1172515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that  the joint density function of  the random variables X and Y is given by <BR>\n" );
document.write( "         F(x,y) ={8xy if 0≤y≤x≤1<BR>\n" );
document.write( "                   0     elsewhere<BR>\n" );
document.write( "a)ComputeP(X+Y <1).(Sketch the region clearly.)<BR>\n" );
document.write( "b)Find the marginal density of X, i.e.fX(x) and marginal density of Y, i.e.fY(y).<BR>\n" );
document.write( "c)Find the conditional density of Y given X=1/2, that is,fY|X(y|1/2).<BR>\n" );
document.write( "d)Find the conditional expectation of Y given X=1/2, that is,E[Y|X=1/2]. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850813 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=850813\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Joint density function:<BR>\n" );
document.write( "    * F(x, y) = 8xy if 0 ≤ y ≤ x ≤ 1<BR>\n" );
document.write( "    * F(x, y) = 0 elsewhere\r<BR>\n" );
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document.write( "**a) Compute P(X + Y < 1)**\r<BR>\n" );
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document.write( "1.  **Sketch the Region:**<BR>\n" );
document.write( "    * The region defined by 0 ≤ y ≤ x ≤ 1 is a triangle in the first quadrant with vertices (0, 0), (1, 0), and (1, 1).<BR>\n" );
document.write( "    * The condition X + Y < 1 implies Y < 1 - X. This is the region below the line Y = 1 - X.<BR>\n" );
document.write( "    * We need to find the intersection of these two regions.<BR>\n" );
document.write( "    * The intersection is a triangle with vertices (0, 0), (1, 0), and (1/2, 1/2).\r<BR>\n" );
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document.write( "2.  **Set up the Integral:**<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫∫ F(x, y) dy dx<BR>\n" );
document.write( "    * The limits of integration are:<BR>\n" );
document.write( "        * 0 ≤ x ≤ 1/2<BR>\n" );
document.write( "        * 0 ≤ y ≤ 1 - x\r<BR>\n" );
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document.write( "3.  **Evaluate the Integral:**<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫(from 0 to 1/2) ∫(from 0 to 1-x) 8xy dy dx<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫(from 0 to 1/2) [4xy^2](from 0 to 1-x) dx<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫(from 0 to 1/2) 4x(1 - x)^2 dx<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫(from 0 to 1/2) 4x(1 - 2x + x^2) dx<BR>\n" );
document.write( "    * P(X + Y < 1) = ∫(from 0 to 1/2) (4x - 8x^2 + 4x^3) dx<BR>\n" );
document.write( "    * P(X + Y < 1) = [2x^2 - (8/3)x^3 + x^4](from 0 to 1/2)<BR>\n" );
document.write( "    * P(X + Y < 1) = 2(1/4) - (8/3)(1/8) + (1/16)<BR>\n" );
document.write( "    * P(X + Y < 1) = 1/2 - 1/3 + 1/16<BR>\n" );
document.write( "    * P(X + Y < 1) = (24 - 16 + 3)/48 = 11/48\r<BR>\n" );
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document.write( "**b) Find the Marginal Densities fX(x) and fY(y)**\r<BR>\n" );
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document.write( "1.  **Marginal Density of X (fX(x)):**<BR>\n" );
document.write( "    * fX(x) = ∫(from 0 to x) 8xy dy<BR>\n" );
document.write( "    * fX(x) = [4xy^2](from 0 to x)<BR>\n" );
document.write( "    * fX(x) = 4x^3, 0 ≤ x ≤ 1\r<BR>\n" );
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document.write( "2.  **Marginal Density of Y (fY(y)):**<BR>\n" );
document.write( "    * fY(y) = ∫(from y to 1) 8xy dx<BR>\n" );
document.write( "    * fY(y) = [4x^2y](from y to 1)<BR>\n" );
document.write( "    * fY(y) = 4y - 4y^3, 0 ≤ y ≤ 1\r<BR>\n" );
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document.write( "**c) Find the Conditional Density fY|X(y|1/2)**\r<BR>\n" );
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document.write( "1.  **Conditional Density Formula:**<BR>\n" );
document.write( "    * fY|X(y|x) = F(x, y) / fX(x)\r<BR>\n" );
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document.write( "2.  **Substitute x = 1/2:**<BR>\n" );
document.write( "    * fY|X(y|1/2) = 8(1/2)y / 4(1/2)^3<BR>\n" );
document.write( "    * fY|X(y|1/2) = 4y/(1/2)<BR>\n" );
document.write( "    * fY|X(y|1/2) = 8y.<BR>\n" );
document.write( "    * Since 0 <= y <= x, and x = 1/2, then 0<=y<=1/2.\r<BR>\n" );
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document.write( "3.  **Result:**<BR>\n" );
document.write( "    * fY|X(y|1/2) = 8y, 0 ≤ y ≤ 1/2\r<BR>\n" );
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document.write( "**d) Find the Conditional Expectation E[Y|X=1/2]**\r<BR>\n" );
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document.write( "1.  **Conditional Expectation Formula:**<BR>\n" );
document.write( "    * E[Y|X=1/2] = ∫ y * fY|X(y|1/2) dy\r<BR>\n" );
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document.write( "2.  **Substitute and Evaluate:**<BR>\n" );
document.write( "    * E[Y|X=1/2] = ∫(from 0 to 1/2) y * 8y dy<BR>\n" );
document.write( "    * E[Y|X=1/2] = ∫(from 0 to 1/2) 8y^2 dy<BR>\n" );
document.write( "    * E[Y|X=1/2] = [(8/3)y^3](from 0 to 1/2)<BR>\n" );
document.write( "    * E[Y|X=1/2] = (8/3) * (1/8)<BR>\n" );
document.write( "    * E[Y|X=1/2] = 1/3\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* a) P(X + Y < 1) = 11/48<BR>\n" );
document.write( "* b) fX(x) = 4x^3, 0 ≤ x ≤ 1; fY(y) = 4y - 4y^3, 0 ≤ y ≤ 1<BR>\n" );
document.write( "* c) fY|X(y|1/2) = 8y, 0 ≤ y ≤ 1/2<BR>\n" );
document.write( "* d) E[Y|X=1/2] = 1/3<BR>\n" );
document.write( " </SPAN>\n" );
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