document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184022>Question 1184022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An insurance policy is written to cover a loss, X, where X has a uniform distribution on<BR>\n" );
document.write( "[0, 1000]. At what level must a deductible be set in order for the expected payment to be<BR>\n" );
document.write( "25% of what it would be with no deductible? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849886 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=849886\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the deductible level:\r<BR>\n" );
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document.write( "**1. Expected Payment with No Deductible:**\r<BR>\n" );
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document.write( "If there's no deductible, the expected payment is simply the expected value of the loss X. For a uniform distribution on [a, b], the expected value is (a+b)/2.  In this case, a = 0 and b = 1000, so:\r<BR>\n" );
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document.write( "E[X] = (0 + 1000) / 2 = 500\r<BR>\n" );
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document.write( "**2. Desired Expected Payment:**\r<BR>\n" );
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document.write( "The problem states that the expected payment with the deductible should be 25% of the expected payment with no deductible:\r<BR>\n" );
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document.write( "Desired Expected Payment = 0.25 * 500 = 125\r<BR>\n" );
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document.write( "**3.  Expected Payment with a Deductible:**\r<BR>\n" );
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document.write( "Let 'd' be the deductible.  The payment Y made by the insurance company is:\r<BR>\n" );
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document.write( "* Y = 0, if X ≤ d<BR>\n" );
document.write( "* Y = X - d, if X > d\r<BR>\n" );
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document.write( "The expected payment with the deductible is:\r<BR>\n" );
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document.write( "E[Y] = ∫ (x - d) * f(x) dx   (integrated from d to 1000)\r<BR>\n" );
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document.write( "Where f(x) is the probability density function of the uniform distribution, which is 1/(1000 - 0) = 1/1000 for 0 ≤ x ≤ 1000.\r<BR>\n" );
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document.write( "So,\r<BR>\n" );
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document.write( "E[Y] = ∫ (x - d) * (1/1000) dx (integrated from d to 1000)<BR>\n" );
document.write( "E[Y] = (1/1000) * [x²/2 - dx] (evaluated from d to 1000)<BR>\n" );
document.write( "E[Y] = (1/1000) * [(1000²/2 - 1000d) - (d²/2 - d²)]<BR>\n" );
document.write( "E[Y] = (1/1000) * [500000 - 1000d - d²/2 + d²]<BR>\n" );
document.write( "E[Y] = (1/1000) * [500000 - 1000d + d²/2]<BR>\n" );
document.write( "E[Y] = 500 - d + d²/2000\r<BR>\n" );
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document.write( "**4. Solve for the Deductible (d):**\r<BR>\n" );
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document.write( "We want E[Y] to equal 125:\r<BR>\n" );
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document.write( "125 = 500 - d + d²/2000<BR>\n" );
document.write( "0 = 375 - d + d²/2000\r<BR>\n" );
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document.write( "Multiply by 2000 to get rid of the fraction:\r<BR>\n" );
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document.write( "0 = 750000 - 2000d + d²\r<BR>\n" );
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document.write( "This is a quadratic equation. Rearrange it into standard form:\r<BR>\n" );
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document.write( "d² - 2000d + 750000 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for d:\r<BR>\n" );
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document.write( "d = (-b ± √(b² - 4ac)) / 2a<BR>\n" );
document.write( "d = (2000 ± √((-2000)² - 4 * 1 * 750000)) / 2<BR>\n" );
document.write( "d = (2000 ± √(4000000 - 3000000)) / 2<BR>\n" );
document.write( "d = (2000 ± √1000000) / 2<BR>\n" );
document.write( "d = (2000 ± 1000) / 2\r<BR>\n" );
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document.write( "This gives us two possible solutions for d:\r<BR>\n" );
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document.write( "* d = (2000 + 1000) / 2 = 1500<BR>\n" );
document.write( "* d = (2000 - 1000) / 2 = 500\r<BR>\n" );
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document.write( "Since the deductible cannot be greater than the maximum possible loss (1000), the correct solution is:\r<BR>\n" );
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document.write( "d = 500\r<BR>\n" );
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document.write( "**Answer:** The deductible must be set at 500.<BR>\n" );
document.write( " </SPAN>\n" );
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