document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206200>Question 1206200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> B. Do the following.\r<BR>\n" );
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document.write( "1. Find the variance and standard deviation of the probability distribution of the X which can take only the values 1, 2, and 3, given that P(1)=\frac{10}{33}, P(2)=\frac{1}{3}, and P(3)=\frac{12}{33}.\r<BR>\n" );
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document.write( "2. Find the variance and standard deviation of the probability distribution of the variable X which can take only the values 3, 5, and 7, given that P(3)=\frac{7}{30}, P(5)=\frac{1}{3}, and P(7)=\frac{13}{30}. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843670 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=843670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll focus on problem 1 only. <BR>\n" );
document.write( "More narrowly, I'll focus on the variance only of this problem.\r<BR>\n" );
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document.write( "Many (online) calculators will quickly compute the variance of a discrete probability distribution. There's not much to explain other than \"type in the numbers, push the right buttons, and out comes the answer\". \r<BR>\n" );
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document.write( "But I'll show what is going on underneath the hood (i.e. how the calculator is computing the variance).\r<BR>\n" );
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document.write( "This is what the table looks like to compute the variance. The approximate variance is shown in <font color=red>red</font><BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD><TD >(X-mu)^2*P(X)</TD></tr><tr><TD >1</TD><TD >0.3030303</TD><TD >0.3030303</TD><TD >0.34087429</TD></tr><tr><TD >2</TD><TD >0.33333333</TD><TD >0.66666666</TD><TD >0.00122436</TD></tr><tr><TD >3</TD><TD >0.36363636</TD><TD >1.09090908</TD><TD >0.32089491</TD></tr><tr><TD ></TD><TD >Sum</TD><TD >2.06060604</TD><TD ><font color=red>0.66299356</font></TD></tr>\n" );
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document.write( "mu = mean = expected value = E[X]<BR>\n" );
document.write( "mu = sum of the X*P(X) values<BR>\n" );
document.write( "mu = 2.06060604 approximately\r<BR>\n" );
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document.write( "The table was generated using LibreOffice spreadsheet (free spreadsheet software). <BR>\n" );
document.write( "If you have Excel then use that instead. Another alternative would be GoogleSheets. <BR>\n" );
document.write( "There are many alternatives to choose from. Feel free to search out your favorite.\r<BR>\n" );
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document.write( "The labels X*P(X) and (X-mu)^2*P(X) at the top of each column explains what is going on to each. If you're confused about this notation then please let me know. \r<BR>\n" );
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document.write( "Unfortunately the table may have some slight rounding error. \r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Take notice that I used the approximate decimal form of each fraction (eg: 10/33 = 0.3030303). \r<BR>\n" );
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document.write( "The reasoning should be fairly obvious once we attempt to find the variance in terms of exact fractions only. Refer to the table below.<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD><TD >(X-mu)^2*P(X)</TD></tr><tr><TD >1</TD><TD >10/33</TD><TD >10/33</TD><TD >12250/35937</TD></tr><tr><TD >2</TD><TD >1/3</TD><TD >2/3</TD><TD >4/3267</TD></tr><tr><TD >3</TD><TD >12/33</TD><TD >12/11</TD><TD >3844/11979</TD></tr><tr><TD ></TD><TD >Sum</TD><TD >68/33</TD><TD ><font color=red>722/1089</font></TD></tr>\n" );
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document.write( "<font color=red>variance = 722/1089 = 0.66299356 approximately</font>.\r<BR>\n" );
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document.write( "The fractions are very large and very messy. <BR>\n" );
document.write( "The table was generated using a custom Python script tailored to find the variance in terms of exact fractions. <BR>\n" );
document.write( "I couldn't find an online solver that would do it in terms of exact fractions, but one may be out there.\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "If your teacher is realistic, then s/he will hopefully accept the approximate version instead of the exact fraction form.<BR>\n" );
document.write( "Sometimes fractions are too much of a pain to bother with.\r<BR>\n" );
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document.write( "Tip: Once you know the variance, apply the square root to find the standard deviation.<BR>\n" );
document.write( "For example, if the variance is 49, then the standard deviation is 7\r<BR>\n" );
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document.write( "For more practice, refer to part (c) in this question <BR>\n" );
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