document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204803>Question 1204803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During my office hours (12pm to 6pm), an average of 2 students per hour comes in for help. Assuming that the probability of a student coming in is uniform throughout my office hours, what are the odds that on a particular day I would have 6 students come in for help? What is the mean number of students that will come see me on a particular day? Variance? Standard deviation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847981 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Random Variable**\r<BR>\n" );
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document.write( "* Let X be the number of students who come in for help during your office hours.\r<BR>\n" );
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document.write( "**2. Determine the Distribution**\r<BR>\n" );
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document.write( "* Since the students arrive at a constant average rate within a given interval (your office hours) and the arrivals are independent, X follows a **Poisson distribution**.\r<BR>\n" );
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document.write( "**3. Calculate the Probability of 6 Students**\r<BR>\n" );
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document.write( "* The Poisson distribution has a single parameter, λ, which represents the average number of occurrences within the given interval. <BR>\n" );
document.write( "    * In this case, λ = 2 students/hour * 6 hours = 12 students/day\r<BR>\n" );
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document.write( "* The probability mass function (PMF) of a Poisson distribution is:<BR>\n" );
document.write( "   P(X = k) = (λ^k * e^(-λ)) / k! <BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "       * k is the number of occurrences (in this case, the number of students)<BR>\n" );
document.write( "       * λ is the average number of occurrences<BR>\n" );
document.write( "       * e is the base of the natural logarithm (approximately 2.71828)\r<BR>\n" );
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document.write( "* To find the probability of 6 students:<BR>\n" );
document.write( "   P(X = 6) = (12^6 * e^(-12)) / 6! <BR>\n" );
document.write( "   P(X = 6) ≈ 0.0113 \r<BR>\n" );
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document.write( "   Therefore, the odds of having 6 students come in for help on a particular day are approximately 0.0113.\r<BR>\n" );
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document.write( "**4. Calculate Mean, Variance, and Standard Deviation**\r<BR>\n" );
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document.write( "* For a Poisson distribution:<BR>\n" );
document.write( "    * Mean (μ) = λ = 12 students/day<BR>\n" );
document.write( "    * Variance (σ²) = λ = 12 students²/day<BR>\n" );
document.write( "    * Standard Deviation (σ) = √λ = √12 ≈ 3.46 students/day\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The probability of having 6 students come in for help is approximately 0.0113.<BR>\n" );
document.write( "* The mean number of students is 12 per day.<BR>\n" );
document.write( "* The variance is 12 students²/day.<BR>\n" );
document.write( "* The standard deviation is approximately 3.46 students/day.<BR>\n" );
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