document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1082961>Question 1082961</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "f(x)= x when 0 < x <= 1<BR>\n" );
document.write( "      2-x when 1 < x <=2\r<BR>\n" );
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document.write( "find out CDF of it </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #696985 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=696985\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given piecewise function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"https://i.imgur.com/bUg2uou.png\"><BR>\n" );
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document.write( "To find the CDF function F(x), we integrate each piece of the piecewise function f(x).<BR>\n" );
document.write( "For the first piece:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{0}^{k} x dx = \left.\frac{x^2}{2}+C\right|_{0}^{k}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{0}^{k} x dx = \left(\frac{k^2}{2}+C\right)-\left(\frac{0^2}{2}+C\right)\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{0}^{k} x dx = \frac{k^2}{2}\"><BR>\n" );
document.write( "where 0 < k <= 1<BR>\n" );
document.write( "So the first piece of F(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%2F2 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 0 < x <= 1.<BR>\n" );
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document.write( "Repeat for the other piece<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{1}^{m} (2-x) dx = \left.2x - \frac{x^2}{2} + C\right|_{1}^{m}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{1}^{m} (2-x) dx = \left(2m - \frac{m^2}{2} + C\right)-\left(2(1) - \frac{1^2}{2} + C\right)\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \int_{1}^{m} (2-x) dx = -\frac{m^2}{2} + 2m-\frac{3}{2}\"><BR>\n" );
document.write( "where 1 < m <= 2<BR>\n" );
document.write( "Since the area is cumulative, we need to include the green shaded triangle (shown below) that has area 1*1/2 = 1/2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/mTSUXGn.png\"><BR>\n" );
document.write( "This green triangle area is represented by F(x) when x = 1.<BR>\n" );
document.write( "It comes from the first part of the piecewise function F(x)<BR>\n" );
document.write( "Add this on to the integral result to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28previous_result%29%2B1%2F2+=+%28%28-m%5E2%29%2F2+%2B+2m+-+3%2F2%29+%2B+1%2F2+=+%28-m%5E2%29%2F2+%2B+2m+-+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "making the second piece of F(x) to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E2%29%2F2+%2B+2x+-+1 ALIGN=MIDDLE  ALT=\"%28-x%5E2%29%2F2+%2B+2x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > when 1 < x <= 2.<BR>\n" );
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document.write( "So overall, the CDF function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"https://i.imgur.com/w2DoDBp.png\"><BR>\n" );
document.write( "which is the final answer<BR>\n" );
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document.write( "Extra Information:<BR>\n" );
document.write( "Graph of the piecwise function f(x) in red, and the piecewise CDF function F(x) in blue<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"https://i.imgur.com/lGwmHRU.png\"><BR>\n" );
document.write( "Image generated by <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> (free graphing software).\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "As you can see, f(x) forms triangle. The base is 2 and the height is 1. The area under all of f(x) is therefore<BR>\n" );
document.write( "A = base*height/2<BR>\n" );
document.write( "A = 2*1/2<BR>\n" );
document.write( "A = 2/2<BR>\n" );
document.write( "A = 1<BR>\n" );
document.write( "Since the area under the function curve f(x) is 1, this confirms that f(x) is indeed a Probability Distribution Function (PDF). </SPAN>\n" );
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