document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206570>Question 1206570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the probability distribution for the random variable x to answer the question.<BR>\n" );
document.write( "x	0	1	2	3	4	5<BR>\n" );
document.write( "p(x)<BR>\n" );
document.write( "0.25	0.05	0.15	0.2	0.05	0.3<BR>\n" );
document.write( "Find 𝜇, <BR>\n" );
document.write( "𝜎2,<BR>\n" );
document.write( " and 𝜎. (Round your standard deviation to two decimal places.)<BR>\n" );
document.write( "𝜇	=	\r<BR>\n" );
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document.write( "𝜎2	=	\r<BR>\n" );
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document.write( "𝜎	=	\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844133 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=844133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "𝜇 = mu<BR>\n" );
document.write( "𝜎 = sigma\r<BR>\n" );
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document.write( "Given table<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >0</TD><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD></tr><tr><TD >p(x)</TD><TD >0.25</TD><TD >0.05</TD><TD >0.15</TD><TD >0.2</TD><TD >0.05</TD><TD >0.3</TD></tr>\n" );
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document.write( "Introduce a new row called x*p(x), where you multiply the paired x and p(x) values.<BR>\n" );
document.write( "Spreadsheet software is recommended.<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >0</TD><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD></tr><tr><TD >p(x)</TD><TD >0.25</TD><TD >0.05</TD><TD >0.15</TD><TD >0.2</TD><TD >0.05</TD><TD >0.3</TD></tr><tr><TD >x*p(x)</TD><TD >0</TD><TD >0.05</TD><TD >0.3</TD><TD >0.6</TD><TD >0.2</TD><TD >1.5</TD></tr>\n" );
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document.write( "Add up the x*p(x) values to get the expected value.<BR>\n" );
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 = 0+0.05+0.3+0.6+0.2+1.5 <BR>\n" );
document.write( "mu = <font color=red>2.65</font>\r<BR>\n" );
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document.write( "We'll use that value of mu to determine the variance, and by extension, the standard deviation as well.\r<BR>\n" );
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document.write( "Introduce a new row called (x-mu)^2*p(x)<BR>\n" );
document.write( "The naming should be self-explanatory. If not then please let me know.<BR>\n" );
document.write( "Example calculation: if x = 0, then (x-mu)^2*p(x) = (0-2.65)^2*0.25 = 1.755625<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >0</TD><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD><TD >5</TD></tr><tr><TD >p(x)</TD><TD >0.25</TD><TD >0.05</TD><TD >0.15</TD><TD >0.2</TD><TD >0.05</TD><TD >0.3</TD></tr><tr><TD >x*p(x)</TD><TD >0</TD><TD >0.05</TD><TD >0.3</TD><TD >0.6</TD><TD >0.2</TD><TD >1.5</TD></tr><tr><TD >(x-mu)^2*p(x)</TD><TD >1.755625</TD><TD >0.136125</TD><TD >0.063375</TD><TD >0.0245</TD><TD >0.091125</TD><TD >1.65675</TD></tr>\n" );
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document.write( "sigma^2 = variance<BR>\n" );
document.write( "sigma^2 = sum of the (x-mu)^2*p(x) values<BR>\n" );
document.write( "sigma^2 = 1.755625+0.136125+0.063375+0.0245+0.091125+1.65675 <BR>\n" );
document.write( "sigma^2 = <font color=red>3.7275</font>\r<BR>\n" );
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document.write( "Side note: another way to find the variance is to compute E[X^2] - (E[X])^2 aka E[X^2] - mu^2<BR>\n" );
document.write( "I'll leave this as an exercise to the reader.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "sigma = standard deviation<BR>\n" );
document.write( "sigma = sqrt( variance )<BR>\n" );
document.write( "sigma = sqrt( 3.7275 )<BR>\n" );
document.write( "sigma = 1.9306735 approximately<BR>\n" );
document.write( "When rounding to 2 decimal places we get <font color=red>1.93</font>\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "mu = <font color=red>2.65</font><BR>\n" );
document.write( "sigma^2 = <font color=red>3.7275</font><BR>\n" );
document.write( "sigma = <font color=red>1.93</font><BR>\n" );
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