document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174329>Question 1174329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> From  past  claims  data  for  a  particular  insurance  business,  an  insurance  company considers  that  claims  for  the  coming  year  will  have  a  mean  size  of  RM5000  and standard  deviation  of  RM7500.  Claim  sizes  are  assumed  to  have  a  lognormal distribution. Estimate the probability of claims exceeding RM20000.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850696 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=850696\">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. Understand Lognormal Distribution**\r<BR>\n" );
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document.write( "* A random variable X has a lognormal distribution if Y = ln(X) has a normal distribution.<BR>\n" );
document.write( "* If X ~ Lognormal(μ, σ²), then ln(X) ~ Normal(μ, σ²).<BR>\n" );
document.write( "* We're given the mean and standard deviation of X (claim sizes), but we need to find the parameters (μ and σ) of the corresponding normal distribution of ln(X).\r<BR>\n" );
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document.write( "**2. Relate Mean and Variance of Lognormal to Normal Parameters**\r<BR>\n" );
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document.write( "* Let E[X] = 5000 and SD[X] = 7500.<BR>\n" );
document.write( "* Let Y = ln(X). Then Y ~ Normal(μ, σ²).<BR>\n" );
document.write( "* We have:<BR>\n" );
document.write( "    * E[X] = exp(μ + σ²/2)<BR>\n" );
document.write( "    * Var[X] = exp(2μ + σ²)(exp(σ²) - 1)\r<BR>\n" );
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document.write( "**3. Solve for μ and σ**\r<BR>\n" );
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document.write( "* E[X] = 5000 = exp(μ + σ²/2)<BR>\n" );
document.write( "* Var[X] = 7500² = 56250000 = exp(2μ + σ²)(exp(σ²) - 1)\r<BR>\n" );
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document.write( "Let's take the natural logarithm of E[X]:\r<BR>\n" );
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document.write( "* ln(5000) = μ + σ²/2<BR>\n" );
document.write( "* 8.517193 = μ + σ²/2<BR>\n" );
document.write( "* μ = 8.517193 - σ²/2\r<BR>\n" );
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document.write( "Now, substitute this into the variance equation:\r<BR>\n" );
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document.write( "* 56250000 = exp(2(8.517193 - σ²/2) + σ²)(exp(σ²) - 1)<BR>\n" );
document.write( "* 56250000 = exp(17.034386 - σ² + σ²)(exp(σ²) - 1)<BR>\n" );
document.write( "* 56250000 = exp(17.034386)(exp(σ²) - 1)<BR>\n" );
document.write( "* 56250000 / exp(17.034386) = exp(σ²) - 1<BR>\n" );
document.write( "* 56250000 / 25000000 = exp(σ²) - 1<BR>\n" );
document.write( "* 2.25 = exp(σ²) - 1<BR>\n" );
document.write( "* 3.25 = exp(σ²)<BR>\n" );
document.write( "* σ² = ln(3.25) ≈ 1.178655<BR>\n" );
document.write( "* σ ≈ √1.178655 ≈ 1.085658\r<BR>\n" );
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document.write( "Now, find μ:\r<BR>\n" );
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document.write( "* μ = 8.517193 - σ²/2 ≈ 8.517193 - 1.178655 / 2 ≈ 8.517193 - 0.5893275 ≈ 7.9278655\r<BR>\n" );
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document.write( "**4. Calculate the Probability**\r<BR>\n" );
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document.write( "* We want to find P(X > 20000).<BR>\n" );
document.write( "* This is equivalent to P(ln(X) > ln(20000)).<BR>\n" );
document.write( "* ln(20000) ≈ 9.903488<BR>\n" );
document.write( "* Let Y = ln(X). We want to find P(Y > 9.903488).<BR>\n" );
document.write( "* Y ~ Normal(7.9278655, 1.178655)<BR>\n" );
document.write( "* Z = (Y - μ) / σ = (9.903488 - 7.9278655) / 1.085658 ≈ 1.8198<BR>\n" );
document.write( "* P(Y > 9.903488) = P(Z > 1.8198)\r<BR>\n" );
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document.write( "Using a standard normal table or calculator:\r<BR>\n" );
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document.write( "* P(Z > 1.8198) ≈ 1 - P(Z ≤ 1.8198) ≈ 1 - 0.9656 ≈ 0.0344\r<BR>\n" );
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document.write( "**Therefore, the estimated probability of claims exceeding RM20000 is approximately 0.0344.**<BR>\n" );
document.write( " </SPAN>\n" );
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