document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1159943>Question 1159943</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The probability of x is given below<BR>\n" );
document.write( "X 		P(x)<BR>\n" );
document.write( "2		.10<BR>\n" );
document.write( "4		.20<BR>\n" );
document.write( "6		.30<BR>\n" );
document.write( "8		.40 <BR>\n" );
document.write( "Answer the following questions<BR>\n" );
document.write( "a)	What is the probability that x is less than or equal to 4?<BR>\n" );
document.write( "b)	What is the probability that x is greater than or equal to 2?<BR>\n" );
document.write( "c)	Calculate the expected value<BR>\n" );
document.write( "d)	Calculate the variance<BR>\n" );
document.write( "e)	Calculate the standard deviation \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 #783079 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=783079\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part a)\r<BR>\n" );
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document.write( "<font color=red size=4>The answer is 0.3</font> since we add the P(x) values when x is 4 or smaller, so when x = 2 or x = 4.\r<BR>\n" );
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document.write( "0.10+0.20 = 0.30 = 0.3\r<BR>\n" );
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document.write( "Part b)\r<BR>\n" );
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document.write( "\"x is greater than or equal to 2\" is basically saying \"every x value mentioned in the table\" because x = 2 is the smallest item listed. <BR>\n" );
document.write( "Add up all the P(x) values. You should get 1 as the result. With any probability distribution, all the P(x) values must add to 1 to represent 100%.\r<BR>\n" );
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document.write( "<font color=red size=4>Answer = 1</font>\r<BR>\n" );
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document.write( "Part c)\r<BR>\n" );
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document.write( "Make a column of the product of the X and P(X) values<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >2</TD><TD >0.1</TD><TD >0.2</TD></tr><tr><TD >4</TD><TD >0.2</TD><TD >0.8</TD></tr><tr><TD >6</TD><TD >0.3</TD><TD >1.8</TD></tr><tr><TD >8</TD><TD >0.4</TD><TD >3.2</TD></tr>\n" );
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document.write( "Then add up the values in that new third column: 0.2+0.8+1.8+3.2 = 6\r<BR>\n" );
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document.write( "<font color=red size=4>Expected value = 6</font>\r<BR>\n" );
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document.write( "The expected value is another term for the mean.\r<BR>\n" );
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document.write( "Part d)\r<BR>\n" );
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document.write( "Make yet another column. This time we'll represent the X^2*P(X) values<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD><TD >X^2*P(X)</TD></tr><tr><TD >2</TD><TD >0.1</TD><TD >0.2</TD><TD >0.4</TD></tr><tr><TD >4</TD><TD >0.2</TD><TD >0.8</TD><TD >3.2</TD></tr><tr><TD >6</TD><TD >0.3</TD><TD >1.8</TD><TD >10.8</TD></tr><tr><TD >8</TD><TD >0.4</TD><TD >3.2</TD><TD >25.6</TD></tr>\n" );
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document.write( "Those new values add to: 0.4+3.2+10.8+25.6 = 40\r<BR>\n" );
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document.write( "Then we subtract off the square of the mean, or expected value, we got back in part c<BR>\n" );
document.write( "40 - (mean)^2 = 40 - 6^2 = 40 - 36 = 4\r<BR>\n" );
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document.write( "<font color=red size=4>The variance is 4</font>\r<BR>\n" );
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document.write( "Part e)\r<BR>\n" );
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document.write( "Apply the square root to the variance to get the standard deviation:<BR>\n" );
document.write( "standard deviation = sqrt(variance)<BR>\n" );
document.write( "standard deviation = sqrt(4)<BR>\n" );
document.write( "<font color=red size=4>standard deviation = 2</font><BR>\n" );
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