document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179730>Question 1179730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X1,...,Xn be a random sample from a distribution with pdf\r<BR>\n" );
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document.write( "f(x, α) = 1 + αx / 2 , −1 ≤ x ≤ 1, and − 1 ≤ α ≤ 1. \r<BR>\n" );
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document.write( "Find the moment estimators for α.\r<BR>\n" );
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document.write( "Thank you... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850178 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the moment estimator for α:\r<BR>\n" );
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document.write( "**1. Find the First Population Moment (E[X]):**\r<BR>\n" );
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document.write( "The first population moment is the expected value (mean) of the distribution:\r<BR>\n" );
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document.write( "E[X] = ∫[-1, 1] x * f(x, α) dx<BR>\n" );
document.write( "E[X] = ∫[-1, 1] x * (1 + αx / 2) dx<BR>\n" );
document.write( "E[X] = ∫[-1, 1] (x + αx² / 2) dx<BR>\n" );
document.write( "E[X] = [x²/2 + αx³/6] evaluated from -1 to 1<BR>\n" );
document.write( "E[X] = [(1/2 + α/6) - (1/2 - α/6)]<BR>\n" );
document.write( "E[X] = α/3\r<BR>\n" );
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document.write( "**2. Find the First Sample Moment (X̄):**\r<BR>\n" );
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document.write( "The first sample moment is the sample mean, denoted as X̄:\r<BR>\n" );
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document.write( "X̄ = (1/n) * Σ[i=1 to n] Xi\r<BR>\n" );
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document.write( "**3. Equate Population and Sample Moments:**\r<BR>\n" );
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document.write( "Set the first population moment equal to the first sample moment:\r<BR>\n" );
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document.write( "E[X] = X̄<BR>\n" );
document.write( "α/3 = X̄\r<BR>\n" );
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document.write( "**4. Solve for α:**\r<BR>\n" );
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document.write( "α = 3 * X̄\r<BR>\n" );
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document.write( "**Therefore, the moment estimator for α is:**\r<BR>\n" );
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document.write( "α̂ = 3 * X̄ = (3/n) * Σ[i=1 to n] Xi<BR>\n" );
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