document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189369>Question 1189369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A discrete random variable x has the following probability distribution:\r<BR>\n" );
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document.write( "x      0 1    2   3<BR>\n" );
document.write( "P(X=x) q 4p^2 p   0.7-4p^2\r<BR>\n" );
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document.write( "a. Find an expression for q in term of p<BR>\n" );
document.write( "b. Find the value of p which gives the largest value of E(X)<BR>\n" );
document.write( "c. Hence find the largest value of E(X) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849272 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849272\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Find an expression for q in terms of p**\r<BR>\n" );
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document.write( "The sum of all probabilities in a probability distribution must equal 1. Therefore,\r<BR>\n" );
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document.write( "q + 4p² + p + 0.7 - 4p² = 1\r<BR>\n" );
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document.write( "q + p + 0.7 = 1\r<BR>\n" );
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document.write( "q = 1 - 0.7 - p\r<BR>\n" );
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document.write( "**q = 0.3 - p**\r<BR>\n" );
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document.write( "**b. Find the value of p which gives the largest value of E(X)**\r<BR>\n" );
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document.write( "The expected value of X, E(X), is given by:\r<BR>\n" );
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document.write( "E(X) = 0*q + 1*(4p²) + 2*p + 3*(0.7 - 4p²)\r<BR>\n" );
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document.write( "E(X) = 4p² + 2p + 2.1 - 12p²\r<BR>\n" );
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document.write( "E(X) = -8p² + 2p + 2.1\r<BR>\n" );
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document.write( "To find the value of p that maximizes E(X), we can take the derivative of E(X) with respect to p and set it to zero.\r<BR>\n" );
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document.write( "d(E(X))/dp = -16p + 2\r<BR>\n" );
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document.write( "Setting the derivative to zero:\r<BR>\n" );
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document.write( "-16p + 2 = 0\r<BR>\n" );
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document.write( "16p = 2\r<BR>\n" );
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document.write( "p = 2/16\r<BR>\n" );
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document.write( "**p = 0.125**\r<BR>\n" );
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document.write( "To confirm that this is a maximum, we can take the second derivative:\r<BR>\n" );
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document.write( "d²(E(X))/dp² = -16\r<BR>\n" );
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document.write( "Since the second derivative is negative, this confirms that p = 0.125 gives a maximum value for E(X).\r<BR>\n" );
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document.write( "**c. Hence find the largest value of E(X)**\r<BR>\n" );
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document.write( "Substitute p = 0.125 into the expression for E(X):\r<BR>\n" );
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document.write( "E(X) = -8(0.125)² + 2(0.125) + 2.1\r<BR>\n" );
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document.write( "E(X) = -8(0.015625) + 0.25 + 2.1\r<BR>\n" );
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document.write( "E(X) = -0.125 + 0.25 + 2.1\r<BR>\n" );
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document.write( "**E(X) = 2.225**\r<BR>\n" );
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document.write( "Therefore, the largest value of E(X) is 2.225. </SPAN>\n" );
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