document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201439>Question 1201439</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of tickets purchased by an individual for Beckham College’s holiday music festival is a uniformly distributed random variable ranging from 6 to 15.<BR>\n" );
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document.write( "Find the mean and standard deviation of this random variable. (Round your answers to 2 decimal places.) </SPAN>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "Mean = <font color=red size=4>10.5</font><BR>\n" );
document.write( "Standard Deviation = <font color=red size=4>2.87</font> (approximate)\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "The number of tickets is a discrete variable. <BR>\n" );
document.write( "It can only be equal to one of the following whole numbers {6,7,8,9,10,11,12,13,14,15}<BR>\n" );
document.write( "This random variable would be continuous if it made sense to have some fractional amount of tickets (eg: 6.785 tickets), but of course it doesn't make sense to have a fractional amount of tickets.\r<BR>\n" );
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document.write( "The mean of a discrete uniform random variable is found by adding up the endpoints, and then dividing in half. <BR>\n" );
document.write( "We are computing the midpoint.<BR>\n" );
document.write( "midpoint = (a+b)/2<BR>\n" );
document.write( "midpoint = (6+15)/2<BR>\n" );
document.write( "midpoint = 21/2<BR>\n" );
document.write( "midpoint = 10.5<BR>\n" );
document.write( "<font color=red>The mean is 10.5</font><BR>\n" );
document.write( "It represents the center of the distribution.<BR>\n" );
document.write( "The proof should be fairly straight-forward, but let me know if you need me to go into more detail in this regard.\r<BR>\n" );
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document.write( "The standard deviation of a discrete uniform random variable isn't as straight-forward.<BR>\n" );
document.write( "The formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%28%28+%28b-a%2B1%29%5E2+-+1+%29%2F12%29 ALIGN=MIDDLE  ALT=\"s+=+sqrt%28%28+%28b-a%2B1%29%5E2+-+1+%29%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "One proof is found here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://proofwiki.org/wiki/Variance_of_Discrete_Uniform_Distribution\">https://proofwiki.org/wiki/Variance_of_Discrete_Uniform_Distribution</a>\r<BR>\n" );
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document.write( "Plug in a = 6 and b = 15<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%28%28+%28b-a%2B1%29%5E2+-+1+%29%2F12%29 ALIGN=MIDDLE  ALT=\"s+=+sqrt%28%28+%28b-a%2B1%29%5E2+-+1+%29%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%28%28+%2815-6%2B1%29%5E2+-+1+%29%2F12%29 ALIGN=MIDDLE  ALT=\"s+=+sqrt%28%28+%2815-6%2B1%29%5E2+-+1+%29%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+2.87228132326901 ALIGN=MIDDLE  ALT=\"s+=+2.87228132326901\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+2.87 ALIGN=MIDDLE  ALT=\"s+=+2.87\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<font color=red>The standard deviation is approximately 2.87</font><BR>\n" );
document.write( "The standard deviation is a measure how spread out a distribution is.\r<BR>\n" );
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document.write( "Side note:<BR>\n" );
document.write( "Some textbooks will present the standard deviation formula as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%28+%28n%5E2-1%29%2F12+%29 ALIGN=MIDDLE  ALT=\"s+=+sqrt%28+%28n%5E2-1%29%2F12+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where n = b-a+1 represents the number of whole numbers from a to b, including both endpoints.<BR>\n" );
document.write( "Example: {6,7,8} has n = b-a+1 = 8-6+1 = 3 items.<BR>\n" );
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