document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203807>Question 1203807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A function f(x) has the following form:<BR>\n" );
document.write( "f(x)=kx^-(k+1)     1 < x < ∞<BR>\n" );
document.write( "and zero otherwise.<BR>\n" );
document.write( "(a)For what values of k is f(x) a pdf?<BR>\n" );
document.write( "(b)Find the CDF based on (a).<BR>\n" );
document.write( "(c)For what values of k does E(X) exist? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845583 by </B><A HREF=/tutors/aboutme.mpl?userid=amarjeeth123><B>amarjeeth123(569)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=amarjeeth123><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=845583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)={kx−(k+1)0​1<x<∞otherwise​\r<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "∫−∞∞f(x)dx=1∫−∞∞​f(x)dx=1\r<BR>\n" );
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document.write( "Then\r<BR>\n" );
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document.write( "k≠0k=0<BR>\n" );
document.write( "∫1∞kx−(k+1)dx=lim⁡A→∞[k−(k+1)+1x−(k+1)+1]A1∫1∞​kx−(k+1)dx=A→∞lim​[−(k+1)+1k​x−(k+1)+1]A1​\r<BR>\n" );
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document.write( "=−lim⁡A→∞A−k+1=1=−A→∞lim​A−k+1=1\r<BR>\n" );
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document.write( "−lim⁡A→∞A−k=0=>k>0−A→∞lim​A−k=0=>k>0\r<BR>\n" );
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document.write( "Answer: k>0k>0\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "F(x)=∫−∞xf(y)dyF(x)=∫−∞x​f(y)dy\r<BR>\n" );
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document.write( "F(x)=0,x<1F(x)=0,x<1\r<BR>\n" );
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document.write( "F(x)=∫1xky−(k+1)dyF(x)=∫1x​ky−(k+1)dy\r<BR>\n" );
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document.write( "=[k−(k+1)+1y−(k+1)+1]x1=−x−k+1=[−(k+1)+1k​y−(k+1)+1]x1​=−x−k+1\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "F(x)={0x<11−x−k1≤x<∞,k>0F(x)={01−x−k​x<11≤x<∞​,k>0\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "E(X)=∫−∞∞xf(x)dx=∫1∞kx−(k+1)+1dxE(X)=∫−∞∞​xf(x)dx=∫1∞​kx−(k+1)+1dx\r<BR>\n" );
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document.write( "=lim⁡A→∞[k−(k+1)+2x−(k+1)+2]A1=A→∞lim​[−(k+1)+2k​x−(k+1)+2]A1​\r<BR>\n" );
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document.write( "=−kk−1lim⁡A→∞A−k+1+kk−1,k>1=−k−1k​A→∞lim​A−k+1+k−1k​,k>1\r<BR>\n" );
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document.write( "If k=1k=1\r<BR>\n" );
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document.write( "E(X)=∫−∞∞xf(x)dx=∫1∞kx−1dxE(X)=∫−∞∞​xf(x)dx=∫1∞​kx−1dx\r<BR>\n" );
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document.write( "=lim⁡A→∞[kln⁡(∣x∣)]A1=does not exist=A→∞lim​[kln(∣x∣)]A1​=does not exist\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( " E(X)E(X) exists for k>1.k>1.  </SPAN>\n" );
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