document.write( "Question 1132181: Let random variable x represent the number of heads when a fair coin is
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\n" ); document.write( " (b) Determine the mean and standard deviation of x. Show all work.
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Algebra.Com's Answer #749122 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let random variable x represent the number of heads when a fair coin is
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document.write( "Before making the table we have to do some other things first.\r\n" );
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document.write( "List the sample space, which is the list of all ways the coins \r\n" );
document.write( "can land:\r\n" );
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document.write( "HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\r\n" );
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document.write( "Each of those are equally likely and since there are 8 of them\r\n" );
document.write( "each has probability 1/8, which is 0.125 as a decimal.\r\n" );
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document.write( "There is only 1 case with 0 heads, TTT, which has probability 1/8.\r\n" );
document.write( "That's the case when X=0 and P(X)=P(0)=1/8=0.125\r\n" );
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document.write( "There are 3 cases with 1 heads, HTT, THT, TTH, So the probability\r\n" );
document.write( "of 1 head is 3 times 1/8 or 3/8 or 0.375 That's the case when X=1 \r\n" );
document.write( "and P(X)=P(1)=3/8=0.375.\r\n" );
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document.write( "There are 3 cases with 2 heads, HHT, HTH, HHT, So the probability\r\n" );
document.write( "of 2 heads is also 3 times 1/8 or 3/8 or 0.375 That's the case \r\n" );
document.write( "when X=2 and P(X)=P(2)=3/8=0.375.\r\n" );
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document.write( "There is only 1 case with 3 heads, HHH, which has probability 1/8.\r\n" );
document.write( "That's the case when X=3 and P(X)=P(3)=1/8=0.125.\r\n" );
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document.write( "Now we can make the table:\r\n" );
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document.write( " X   P(X)\r\n" );
document.write( " --------\r\n" );
document.write( " 0   0.125\r\n" );
document.write( " 1   0.375\r\n" );
document.write( " 2   0.375\r\n" );
document.write( " 3   0.125\r\n" );
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document.write( "That's the answer to (a).\r\n" );
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\n" ); document.write( "(b) Determine the mean m and standard deviation s of x.
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document.write( "We make a new table, starting with the same first column.\r\n" );
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document.write( "  X\r\n" );
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document.write( "  0\r\n" );
document.write( "  1\r\n" );
document.write( "  2\r\n" );
document.write( "  3\r\n" );
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document.write( "Under the bottom line we determine the mean by averaging the 4\r\n" );
document.write( "numbers of heads in all the rolls. We add the column and divide\r\n" );
document.write( "by 4:\r\n" );
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document.write( "  X\r\n" );
document.write( "  -----------------\r\n" );
document.write( "  0\r\n" );
document.write( "  1\r\n" );
document.write( "  2\r\n" );
document.write( "  3\r\n" );
document.write( "  ----------------  \r\n" );
document.write( "4)6.0\r\n" );
document.write( "  1.5\r\n" );
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document.write( "the mean = m = (0+1+2+3)÷4 = 6÷4 = 6/4 = 3/2 = 1.5\r\n" );
document.write( "So the mean = m = 1.5. Next we subtract the mean of 1.5 from\r\n" );
document.write( "each of the values of X:\r\n" );
document.write( "For X=0, we subtract 1.5 from 0 and get X-m=0-1.5 = -1.5\r\n" );
document.write( "For X=1, we subtract 1.5 from 1 and get X-m=1-1.5 = -0.5\r\n" );
document.write( "For X=2, we subtract 1.5 from 2 and get X-m=2-1.5 = +0.5\r\n" );
document.write( "For X=3, we subtract 1.5 from 3 and get X-m=3-1.5 = +1.5\r\n" );
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document.write( "Then we list those in a new column out beside the first:\r\n" );
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document.write( "  X      X-m\r\n" );
document.write( "  -----------------\r\n" );
document.write( "  0     -1.5\r\n" );
document.write( "  1     -0.5\r\n" );
document.write( "  2     +0.5\r\n" );
document.write( "  3     +1.5\r\n" );
document.write( "  ----------------  \r\n" );
document.write( "4)6.0         \r\n" );
document.write( "  1.5\r\n" );
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document.write( "Next we square each of the values of X-m:\r\n" );
document.write( "For X-m = -1.5, we square and get (X-m)²=(-1.5)²= 2.25\r\n" );
document.write( "For X-m = -0.5, we square and get (X-m)²=(-0.5)²= 0.25\r\n" );
document.write( "For X-m = +0.5, we square and get (X-m)²=(+0.5)²= 0.25\r\n" );
document.write( "For X-m = +1.5, we square and get (X-m)²=(+1.5)²= 2.25\r\n" );
document.write( "Then we list those in a new column out beside the second:\r\n" );
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document.write( "  X      X-m     (X-m)²\r\n" );
document.write( "  ---------------------\r\n" );
document.write( "  0     -1.5     2.25  \r\n" );
document.write( "  1     -0.5     0.25\r\n" );
document.write( "  2     +0.5     0.25\r\n" );
document.write( "  3     +1.5     2.25\r\n" );
document.write( "  -------------------  \r\n" );
document.write( "4)6.0                 \r\n" );
document.write( "  1.5\r\n" );
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document.write( "Under the bottom line we determine the variance by averaging the 4\r\n" );
document.write( "values of (X-m)² in all the rolls. We add the column and divide\r\n" );
document.write( "by 4.\r\n" );
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document.write( "NOTICE: We divide by 4 because this is a POPULATION, not a SAMPLE.\r\n" );
document.write( "If it were a SAMPLE, we would divide by 1 less than 4, or 3.  But\r\n" );
document.write( "this is the ENTIRE POPULATION of ALL the ways the coin can land on\r\n" );
document.write( "the three tosses, so we divide by 4, not 3:\r\n" );
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document.write( "  X      X-m     (X-m)²\r\n" );
document.write( "  ---------------------\r\n" );
document.write( "  0     -1.5     2.25  \r\n" );
document.write( "  1     -0.5     0.25\r\n" );
document.write( "  2     +0.5     0.25\r\n" );
document.write( "  3     +1.5     2.25\r\n" );
document.write( "  -------------------  \r\n" );
document.write( "4)6.0          4)5.00         \r\n" );
document.write( "  1.5=m          1.25=s²=the variance\r\n" );
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document.write( "Finally we find the standard deviation, s, by taking the\r\n" );
document.write( "square root of the variance s². The square root of 1.25\r\n" );
document.write( "is approximately 1.118033989, so that's the standard \r\n" );
document.write( "deviation s.  Round that off \r\n" );
document.write( "as your teacher told you.\r\n" );
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document.write( "  X      X-m     (X-m)²\r\n" );
document.write( "  ---------------------\r\n" );
document.write( "  0     -1.5     2.25  \r\n" );
document.write( "  1     -0.5     0.25\r\n" );
document.write( "  2     +0.5     0.25\r\n" );
document.write( "  3     +1.5     2.25\r\n" );
document.write( "  -------------------  \r\n" );
document.write( "4)6.0          4)5.00        \r\n" );
document.write( "  1.5=m          1.25=s²=the variance\r\n" );
document.write( "                 1.118033989=s=the standard deviation\r\n" );
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document.write( "Edwin

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