document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177399>Question 1177399</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A discrete random variable X is such that<BR>\n" );
document.write( "            P(X = n) = 2^n−1 / 3^n , n = 1, 2, . . . , n, . . . .<BR>\n" );
document.write( "Show that EX = 3\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 #850470 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=850470\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define the Expected Value**\r<BR>\n" );
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document.write( "The expected value of a discrete random variable X is defined as:\r<BR>\n" );
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document.write( "* E(X) = Σ [x * P(X = x)]\r<BR>\n" );
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document.write( "In our case, X takes values n, and P(X = n) = 2^(n-1) / 3^n. So:\r<BR>\n" );
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document.write( "* E(X) = Σ [n * (2^(n-1) / 3^n)] for n = 1, 2, 3, ...\r<BR>\n" );
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document.write( "**2. Simplify the Expression**\r<BR>\n" );
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document.write( "* E(X) = Σ [n * (2^(n-1) / 3^n)]<BR>\n" );
document.write( "* E(X) = Σ [n * (1/2) * (2^n / 3^n)]<BR>\n" );
document.write( "* E(X) = (1/2) * Σ [n * (2/3)^n]\r<BR>\n" );
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document.write( "**3. Recognize the Series**\r<BR>\n" );
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document.write( "We have the series:\r<BR>\n" );
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document.write( "* Σ [n * (2/3)^n]\r<BR>\n" );
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document.write( "Let's use the formula for the sum of the series Σ nx^(n-1) = 1/(1-x)^2\r<BR>\n" );
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document.write( "We can rewrite the sum as:\r<BR>\n" );
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document.write( "* Σ [n * (2/3)^n] = (2/3) * Σ [n * (2/3)^(n-1)]\r<BR>\n" );
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document.write( "Now, let x = 2/3. Then:\r<BR>\n" );
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document.write( "* Σ [n * (2/3)^(n-1)] = 1 / (1 - 2/3)^2<BR>\n" );
document.write( "* Σ [n * (2/3)^(n-1)] = 1 / (1/3)^2<BR>\n" );
document.write( "* Σ [n * (2/3)^(n-1)] = 1 / (1/9)<BR>\n" );
document.write( "* Σ [n * (2/3)^(n-1)] = 9\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "* Σ [n * (2/3)^n] = (2/3) * 9 = 6\r<BR>\n" );
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document.write( "**4. Calculate E(X)**\r<BR>\n" );
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document.write( "* E(X) = (1/2) * Σ [n * (2/3)^n]<BR>\n" );
document.write( "* E(X) = (1/2) * 6<BR>\n" );
document.write( "* E(X) = 3\r<BR>\n" );
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document.write( "**Therefore, E(X) = 3.**<BR>\n" );
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