document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204182>Question 1204182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a laboratory assignment, if the equipment is<BR>\n" );
document.write( "working, the density function of the observed outcome<BR>\n" );
document.write( "X is<BR>\n" );
document.write( "f(x) = {2(1 − x), 0 < x < 1,<BR>\n" );
document.write( "       {0, otherwise.<BR>\n" );
document.write( "Find the variance and standard deviation of X. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840280 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=840280\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Review the formulas mentioned here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/4%3A_Continuous_Random_Variables/4.2%3A_Expected_Value_and_Variance_of_Continuous_Random_Variables\">https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/MATH_345__-_Probability_(Kuter)/4%3A_Continuous_Random_Variables/4.2%3A_Expected_Value_and_Variance_of_Continuous_Random_Variables</a>\r<BR>\n" );
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document.write( "E[X] = expected value of X = mean\r<BR>\n" );
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document.write( "The formula on that page says:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(X) = \int_{-\infty}^{\infty}\text{x}f(\text{x})d\text{x}\"><BR>\n" );
document.write( "However, we do not have to worry about the entire real number line.<BR>\n" );
document.write( "Instead, we can shrink the domain to focus on the interval 0 < x < 1, since f(x) = 0 otherwise.\r<BR>\n" );
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document.write( "You need to calculate <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(X) = \int_{0}^{1}\text{x}f(\text{x})d\text{x} = \int_{0}^{1}2\text{x}(1-\text{x})d\text{x}\">\r<BR>\n" );
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document.write( "I'll skip steps<BR>\n" );
document.write( "Integrating should be fairly trivial because 2x(1-x) = 2x-2x^2 is a polynomial\r<BR>\n" );
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document.write( "You should find that<BR>\n" );
document.write( "E[X] = 1/3<BR>\n" );
document.write( "in other words,<BR>\n" );
document.write( "mu = mean = 1/3\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "The variance is defined as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large Var(X) = E(X^2) - \mu^2\"><BR>\n" );
document.write( "i.e.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large Var(X) = E(X^2) - (E(X))^2\">\r<BR>\n" );
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document.write( "So we'll need to calculate E(X^2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(X^2) = \int_{0}^{1}\text{x}^2f(\text{x})d\text{x} = \int_{0}^{1}2\text{x}^2(1-x)d\text{x}\"><BR>\n" );
document.write( "Once again I'll skip steps. \r<BR>\n" );
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document.write( "The result you should get is E[X^2] = 1/6\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "Var(X) = variance<BR>\n" );
document.write( "Var(X) = E[X^2] - ( E[X] )^2<BR>\n" );
document.write( "Var(X) = 1/6 - ( 1/3 )^2<BR>\n" );
document.write( "Var(X) = 1/18\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "SD(X) = standard deviation<BR>\n" );
document.write( "SD(X) = sqrt( Var(X) )<BR>\n" );
document.write( "SD(X) = sqrt( 1/18 )<BR>\n" );
document.write( "SD(X) = 1/sqrt(18)<BR>\n" );
document.write( "SD(X) = sqrt(18)/18<BR>\n" );
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