document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185234>Question 1185234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An insurance company charges a customer an annual premium of $100, and <BR>\n" );
document.write( "there is a probability of 0.9 that the customer will not need to make a claim. If the <BR>\n" );
document.write( "customer does make a claim, the amount of the claim $𝑋 has a probability density <BR>\n" );
document.write( "function<BR>\n" );
document.write( "𝑓(𝑥) = 𝑥(1800 − 𝑥)/972,000,000<BR>\n" );
document.write( "For 0 ≤ 𝑥 ≤ 1800. Each customer also incurs administrative costs to the insurance company <BR>\n" );
document.write( "of $5. If the insurance company has 10,000 customers, what is its expected annual profit? <BR>\n" );
document.write( "Would you expect the customers’ claim to be independent of each other? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849765 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=849765\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the insurance company's expected annual profit:\r<BR>\n" );
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document.write( "**1. Calculate the expected claim amount:**\r<BR>\n" );
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document.write( "The expected claim amount is calculated by integrating the product of the claim amount (x) and the probability density function f(x) over the range of possible claim amounts:\r<BR>\n" );
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document.write( "E[X] = ∫₀¹⁸⁰⁰ x * f(x) dx <BR>\n" );
document.write( "E[X] = ∫₀¹⁸⁰⁰ x * [x(1800 - x) / 972,000,000] dx<BR>\n" );
document.write( "E[X] = (1/972,000,000) ∫₀¹⁸⁰⁰ (1800x² - x³) dx<BR>\n" );
document.write( "E[X] = (1/972,000,000) [600x³ - (x⁴/4)] from 0 to 1800<BR>\n" );
document.write( "E[X] = (1/972,000,000) * [600(1800)³ - (1800)⁴/4]<BR>\n" );
document.write( "E[X] = (1/972,000,000) * 1,944,000,000<BR>\n" );
document.write( "E[X] = $1000\r<BR>\n" );
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document.write( "**2. Calculate the expected profit per customer:**\r<BR>\n" );
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document.write( "*   Premium per customer: $100<BR>\n" );
document.write( "*   Administrative cost per customer: $5<BR>\n" );
document.write( "*   Probability of making a claim: 1 - 0.9 = 0.1<BR>\n" );
document.write( "*   Expected claim amount: $1000\r<BR>\n" );
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document.write( "Expected profit per customer = Premium - Administrative cost - (Probability of claim * Expected claim amount)<BR>\n" );
document.write( "Expected profit per customer = $100 - $5 - (0.1 * $1000)<BR>\n" );
document.write( "Expected profit per customer = $95 - $100<BR>\n" );
document.write( "Expected profit per customer = -$5\r<BR>\n" );
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document.write( "**3. Calculate the expected annual profit for 10,000 customers:**\r<BR>\n" );
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document.write( "Expected annual profit = Expected profit per customer * Number of customers<BR>\n" );
document.write( "Expected annual profit = -$5 * 10,000<BR>\n" );
document.write( "Expected annual profit = -$50,000\r<BR>\n" );
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document.write( "**4. Independence of claims:**\r<BR>\n" );
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document.write( "It's reasonable to assume that customers' claims are independent of each other.  One customer making a claim should not generally affect the probability of another customer making a claim.  There might be some rare exceptions (e.g., a widespread natural disaster causing many claims), but for the vast majority of individual claims, independence is a valid assumption.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "The insurance company's expected annual profit is -$50,000. This means that, on average, the company can expect to lose $50,000 per year with the current premium structure.  The assumption of independence between customer claims is generally reasonable.<BR>\n" );
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