document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088788>Question 1088788</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sports Score Hot Line Calls (Optional) Sports Scores Hot Line receives, on the average, 8 calls per hour requesting the latest sports scores. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive a. At least 8 calls b. 3 or more calls c. At most 7 calls. Full solution please </SPAN>\n" );
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document.write( "Sports Score Hot Line Calls (Optional) Sports Scores Hot Line receives, on the average, 8 calls per hour requesting the latest sports scores. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive a. At least 8 calls b. 3 or more calls c. At most 7 calls. Full solution please<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Given it is a Poisson process, describing number of occurrences within a given period of time.  The situation fits the requirements of the Poisson distribution.\r<BR>\n" );
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document.write( "The PMF of the distribution is given by:<BR>\n" );
document.write( "P(k)=lambda^k*e^(-lambda)/k! ..........(1)<BR>\n" );
document.write( "where lambda is the mean number of occurrences within a given time period<BR>\n" );
document.write( "The CDF exists in closed form that involves gamma functions.  We could get by using only PMF.\r<BR>\n" );
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document.write( "Here we have lambda=8 calls per hour (period is 1 hour)<BR>\n" );
document.write( "Using equation (1) above, we calculate P(k) for k=0 to 8, using lambda=8:<BR>\n" );
document.write( "Example: k=0, P(0)=8^0*e^-8/0!=0.000335<BR>\n" );
document.write( "k P(k)<BR>\n" );
document.write( "0 0.000335<BR>\n" );
document.write( "1 0.002683<BR>\n" );
document.write( "2 0.010735<BR>\n" );
document.write( "3 0.028626<BR>\n" );
document.write( "4 0.057252<BR>\n" );
document.write( "5 0.091604<BR>\n" );
document.write( "6 0.122138<BR>\n" );
document.write( "7 0.139587<BR>\n" );
document.write( "8 0.139587\r<BR>\n" );
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document.write( "Total=0.592547<BR>\n" );
document.write( " <BR>\n" );
document.write( "With the above table, we can then answer the given questions.<BR>\n" );
document.write( "(a) At least 8 calls<BR>\n" );
document.write( "P(K>=8)=1-P(K<8)<BR>\n" );
document.write( "=1-(P(0)+P(1)+P(2)...+P(7))<BR>\n" );
document.write( "=1-(0.000335+0.002683+0.010735+0.028626+0.057252+0.091604+0.122138+0.139587)<BR>\n" );
document.write( "=1-0.452961<BR>\n" );
document.write( "=0.547039<BR>\n" );
document.write( "  <BR>\n" );
document.write( "(b) Three or more calls<BR>\n" );
document.write( "P(K>=3)=1-P(K<3)<BR>\n" );
document.write( "=1-(P(0)+P(1)+P(2))<BR>\n" );
document.write( "=1-((0.000335+0.002683+0.010735)<BR>\n" );
document.write( "=0.986246<BR>\n" );
document.write( " <BR>\n" );
document.write( "(c) At most 7 calls<BR>\n" );
document.write( "P(K<=7)=1-P(K>=8)<BR>\n" );
document.write( "=1-0.547039<BR>\n" );
document.write( "=0.452961<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );