document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1087230>Question 1087230</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A fair coin is tossed 6times. The probability distribution of X, the number of heads is recorded in the table below;\r<BR>\n" );
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document.write( "x	0	1	2	3	4	5	6<BR>\n" );
document.write( "P(X=x) 0.16    0.093  0.23    0.46    0.23    0.093   0.16 <BR>\n" );
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document.write( "Find the expected number of heads. \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 #701502 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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/cgi-bin/show-face.mpl?user=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=701502\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "A fair coin is tossed 6times. The probability distribution of X, the number of heads is recorded in the table below;<BR>\n" );
document.write( "x	0	1	2	3	4	5	6<BR>\n" );
document.write( "P(X=x) 0.16    0.093  0.23    0.46    0.23    0.093   0.16 <BR>\n" );
document.write( "Find the expected number of heads. <BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "The expected value is the sum &sum;x*p(x)<BR>\n" );
document.write( "where x is the value of the outcome, and p(x) is the corresponding probability.<BR>\n" );
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document.write( "For the given distribution, first verify that it is in fact is a distribution by adding up the probabilities.  If they do not add up to exactly one, something is amiss.<BR>\n" );
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document.write( "We find that the sum adds up to 1.426, so there is a problem with the given distribution, and work cannot progress unless this is corrected.<BR>\n" );
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document.write( "The correction can be made by recalculating the probabilities using the binomial distribution, the results of which are as follows:<BR>\n" );
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document.write( "x P(X=x) <BR>\n" );
document.write( "0 0.015625<BR>\n" );
document.write( "1 0.09375<BR>\n" );
document.write( "2 0.234375<BR>\n" );
document.write( "3 0.3125<BR>\n" );
document.write( "4 0.234375<BR>\n" );
document.write( "5 0.09375<BR>\n" );
document.write( "6 0.015625<BR>\n" );
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document.write( "From the symmetry, we can tell that the expected value (=mean) is 3.<BR>\n" );
document.write( "Using the above formula, we find<BR>\n" );
document.write( "&sum; x*P(X=x) <BR>\n" );
document.write( "=(0.0+0.09375+0.46875+0.9375+0.9375+0.46875+0.09375)<BR>\n" );
document.write( "=3   as expected.\r<BR>\n" );
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