document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203803>Question 1203803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A discrete random variable has pdf f(x).<BR>\n" );
document.write( "(a) If f(x) = k(1/2)^x for x= 1, 2, 3, and zero otherwise, find k.<BR>\n" );
document.write( "(b) Is a function of the form f(x) = k[(1/2)^x - 1/2] for x = 0, 1, 2 a pdf for any k? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839764 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=839764\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "f(x) = k(1/2)^x<BR>\n" );
document.write( "f(1) = k(1/2)^1<BR>\n" );
document.write( "f(1) = k/2\r<BR>\n" );
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document.write( "f(x) = k(1/2)^x<BR>\n" );
document.write( "f(2) = k(1/2)^2<BR>\n" );
document.write( "f(2) = k/4\r<BR>\n" );
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document.write( "f(x) = k(1/2)^x<BR>\n" );
document.write( "f(3) = k(1/2)^3<BR>\n" );
document.write( "f(3) = k/8\r<BR>\n" );
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document.write( "For any PDF, the f(x) probability values add to 1.\r<BR>\n" );
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document.write( "f(1) + f(2) + f(3) = 1<BR>\n" );
document.write( "k/2 + k/4 + k/8 = 1<BR>\n" );
document.write( "4k/8 + 2k/8 + k/8 = 1<BR>\n" );
document.write( "(4k+2k+k)/8 = 1<BR>\n" );
document.write( "7k/8 = 1<BR>\n" );
document.write( "7k = 1*8<BR>\n" );
document.write( "7k = 8<BR>\n" );
document.write( "<font color=red size=4>k = 8/7</font>\r<BR>\n" );
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document.write( "=========================================\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "f(x) = k[(1/2)^x - 1/2]<BR>\n" );
document.write( "f(0) = k[(1/2)^0 - 1/2]<BR>\n" );
document.write( "f(0) = k/2\r<BR>\n" );
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document.write( "f(x) = k[(1/2)^x - 1/2]<BR>\n" );
document.write( "f(1) = k[(1/2)^1 - 1/2]<BR>\n" );
document.write( "f(1) = 0\r<BR>\n" );
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document.write( "f(x) = k[(1/2)^x - 1/2]<BR>\n" );
document.write( "f(2) = k[(1/2)^2 - 1/2]<BR>\n" );
document.write( "f(2) = -k/4\r<BR>\n" );
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document.write( "f(0)+f(1)+f(2) = 1<BR>\n" );
document.write( "k/2 + 0 + (-k/4) = 1<BR>\n" );
document.write( "2k/4 - k/4 = 1<BR>\n" );
document.write( "(2k-k)/4 = 1<BR>\n" );
document.write( "k/4 = 1<BR>\n" );
document.write( "k = 4\r<BR>\n" );
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document.write( "f(x) is a PDF if and only if <font color=red size=4>k = 4</font><BR>\n" );
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