document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195988>Question 1195988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the number of cars that arrive at a car wash during one hour is described by a Poisson probability distribution with a mean of 6 cars per hour. Now we are interested in the time, denoted by X, between the arrivals. \r<BR>\n" );
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document.write( "1. What is probability distribution of X? \r<BR>\n" );
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document.write( "2. What is probability that arrival time is greater than 10 minutes?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848495 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848495\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Probability Distribution of X**\r<BR>\n" );
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document.write( "* **If the number of arrivals per hour follows a Poisson distribution, then the time between arrivals follows an Exponential distribution.**\r<BR>\n" );
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document.write( "**2. Probability that Arrival Time is Greater Than 10 Minutes**\r<BR>\n" );
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document.write( "* **Convert minutes to hours:** 10 minutes = 10/60 hours = 1/6 hour <BR>\n" );
document.write( "* **Exponential Distribution:**<BR>\n" );
document.write( "    * The probability density function (PDF) of an exponential distribution is: <BR>\n" );
document.write( "        * f(x) = λ * e^(-λx) <BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * λ is the rate parameter (average number of arrivals per unit time) = 6 cars/hour<BR>\n" );
document.write( "            * x is the time between arrivals (in hours)\r<BR>\n" );
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document.write( "* **Cumulative Distribution Function (CDF):**<BR>\n" );
document.write( "    * The probability of the time between arrivals being less than or equal to 't' is given by:<BR>\n" );
document.write( "        * F(t) = 1 - e^(-λt)\r<BR>\n" );
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document.write( "* **Probability of Arrival Time Greater Than 10 Minutes:**<BR>\n" );
document.write( "    * P(X > 1/6) = 1 - P(X ≤ 1/6) <BR>\n" );
document.write( "    * P(X > 1/6) = 1 - F(1/6)<BR>\n" );
document.write( "    * P(X > 1/6) = 1 - (1 - e^(-6 * (1/6)))<BR>\n" );
document.write( "    * P(X > 1/6) = 1 - (1 - e^(-1))<BR>\n" );
document.write( "    * P(X > 1/6) = e^(-1) <BR>\n" );
document.write( "    * P(X > 1/6) ≈ 0.3679\r<BR>\n" );
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document.write( "**Therefore, the probability that the arrival time is greater than 10 minutes is approximately 0.3679 or 36.79%.**<BR>\n" );
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