document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170116>Question 1170116</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random variable X has a poisson distribution with parameter m. If P(X=2)=0.2565, find the value of m  given that e=2.7183 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851248 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=851248\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Alright, let's solve this Poisson distribution problem step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Poisson Distribution**\r<BR>\n" );
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document.write( "The Poisson distribution gives the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The probability mass function (PMF) of a Poisson distribution is:\r<BR>\n" );
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document.write( "$$P(X=k) = \frac{e^{-m} m^k}{k!}$$\r<BR>\n" );
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document.write( "where:\r<BR>\n" );
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document.write( "* X is the random variable representing the number of events.<BR>\n" );
document.write( "* k is the number of events (non-negative integer).<BR>\n" );
document.write( "* m is the average number of events (the parameter).<BR>\n" );
document.write( "* e is Euler's number (approximately 2.7183).\r<BR>\n" );
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document.write( "**Applying the Given Information**\r<BR>\n" );
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document.write( "We are given that:\r<BR>\n" );
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document.write( "* X follows a Poisson distribution with parameter m.<BR>\n" );
document.write( "* P(X=2) = 0.2565\r<BR>\n" );
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document.write( "We need to find the value of m.\r<BR>\n" );
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document.write( "Using the Poisson PMF with k=2:\r<BR>\n" );
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document.write( "$$P(X=2) = \frac{e^{-m} m^2}{2!}$$\r<BR>\n" );
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document.write( "We know P(X=2) = 0.2565, so:\r<BR>\n" );
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document.write( "$$0.2565 = \frac{e^{-m} m^2}{2}$$\r<BR>\n" );
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document.write( "Multiply both sides by 2:\r<BR>\n" );
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document.write( "$$0.513 = e^{-m} m^2$$\r<BR>\n" );
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document.write( "$$0.513 = \frac{m^2}{e^{m}}$$\r<BR>\n" );
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document.write( "Now, we need to solve for m. This equation is transcendental, meaning it can't be solved algebraically. We'll need to use numerical methods or approximations.\r<BR>\n" );
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document.write( "**Solving for m (Numerical Approximation)**\r<BR>\n" );
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document.write( "We'll use a trial-and-error approach, plugging in values of m to see which one gets us closest to 0.513.\r<BR>\n" );
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document.write( "Let's try m = 2:\r<BR>\n" );
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document.write( "$$\frac{2^2}{e^2} = \frac{4}{2.7183^2} \approx \frac{4}{7.389} \approx 0.5413$$\r<BR>\n" );
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document.write( "This is close.\r<BR>\n" );
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document.write( "Let's try m = 1.9:\r<BR>\n" );
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document.write( "$$\frac{1.9^2}{e^{1.9}} = \frac{3.61}{6.6858} \approx 0.540$$\r<BR>\n" );
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document.write( "Let's try m = 2.1:\r<BR>\n" );
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document.write( "$$\frac{2.1^2}{e^{2.1}} = \frac{4.41}{8.1662} \approx 0.540$$\r<BR>\n" );
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document.write( "Let's try m = 1.8:\r<BR>\n" );
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document.write( "$$\frac{1.8^2}{e^{1.8}} = \frac{3.24}{6.0496} \approx 0.5355$$\r<BR>\n" );
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document.write( "Let's try m = 1.6:<BR>\n" );
document.write( "$$\frac{1.6^2}{e^{1.6}} = \frac{2.56}{4.9530} \approx 0.5168$$\r<BR>\n" );
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document.write( "Let's try m = 1.58:<BR>\n" );
document.write( "$$\frac{1.58^2}{e^{1.58}} = \frac{2.4964}{4.8550} \approx 0.5142$$\r<BR>\n" );
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document.write( "Let's try m = 1.57:<BR>\n" );
document.write( "$$\frac{1.57^2}{e^{1.57}} = \frac{2.4649}{4.8066} \approx 0.5128$$\r<BR>\n" );
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document.write( "Therefore, m is approximately 1.57.\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "The value of m is approximately 1.57.<BR>\n" );
document.write( " </SPAN>\n" );
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