document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198958>Question 1198958</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Benford’s law states that in a very large variety of real-life data sets, the first digit approximately follows a particular distribution with about a 30% chance of a 1, an 18% chance of a 2, and in general\r<BR>\n" );
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document.write( "P ( D = j ) = log base 10 of ( j+1/j) , for j element of{ 1,2,3,4,5,6,7,8,9}\r<BR>\n" );
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document.write( "where D is the first digit of a randomly chosen element.\r<BR>\n" );
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document.write( "Check whether this is a valid Probability mass function (PMF) and why. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832646 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=832646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(D = j) = \log_{10}\left(\frac{j+1}{j}\right), \ \ \text{ where } \ j \in \{ 1,2,3,4,5,6,7,8,9\}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(D = 1) = \log_{10}\left(\frac{1+1}{1}\right) = \log_{10}\left(2\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(D = 2) = \log_{10}\left(\frac{2+1}{2}\right) = \log_{10}\left(\frac{3}{2}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(D = 3) = \log_{10}\left(\frac{3+1}{3}\right) = \log_{10}\left(\frac{4}{3}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(D = 4) = \log_{10}\left(\frac{4+1}{4}\right) = \log_{10}\left(\frac{5}{4}\right)\">\r<BR>\n" );
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document.write( "and so on.\r<BR>\n" );
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document.write( "We have these base 10 logs for terms j=1 through j=9<BR>\n" );
document.write( "log(2)<BR>\n" );
document.write( "log(3/2)<BR>\n" );
document.write( "log(4/3)<BR>\n" );
document.write( "log(5/4)<BR>\n" );
document.write( "log(6/5)<BR>\n" );
document.write( "log(7/6)<BR>\n" );
document.write( "log(8/7)<BR>\n" );
document.write( "log(9/8)<BR>\n" );
document.write( "log(10/9)\r<BR>\n" );
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document.write( "Adding up those logs, and then using the rule<BR>\n" );
document.write( "log(A)+log(B) = log(A*B)<BR>\n" );
document.write( "will get us this<BR>\n" );
document.write( "log( 2*(3/2)*(4/3)*(5/4)*(6/5)*(7/6)*(8/7)*(9/8)*(10/9) )\r<BR>\n" );
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document.write( "Each denominator cancels out with the previous numerator. <BR>\n" );
document.write( "This means every denominator goes away, and nearly every numerator also goes away.\r<BR>\n" );
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document.write( "We'll be left with the single term \"10\" inside the log.<BR>\n" );
document.write( "Then we say log(10) = 1\r<BR>\n" );
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document.write( "Therefore, the sum of the probability values is 1, which satisfies one of the requirements for the PMF.\r<BR>\n" );
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document.write( "The other requirement is that each P(j) value is between 0 and 1.<BR>\n" );
document.write( "0 ≤ log(x) ≤ 1<BR>\n" );
document.write( "10^0 ≤ 10^log(x) ≤ 10^1<BR>\n" );
document.write( "1 ≤ x ≤ 10<BR>\n" );
document.write( "To have P(j) between 0 and 1, it is equivalent to having the argument of the log between 1 and 10.<BR>\n" );
document.write( "This applies to each log mentioned, so we have satisfied the second requirement.\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Answer: This is a valid PMF since the P(j) values add to 1, and each P(j) value is between 0 and 1.<BR>\n" );
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