document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164707>Question 1164707</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Five observations have been drawn independently and at random from<BR>\n" );
document.write( "a continuous distribution. What is the probability that the next observation<BR>\n" );
document.write( "will be less than all of the first five? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853525 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=853525\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic problem in **non-parametric statistics** and **symmetry**. Because the observations are drawn from a continuous distribution, the probability that any two observations are exactly equal is zero.\r<BR>\n" );
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document.write( "### The Symmetry Argument\r<BR>\n" );
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document.write( "When you have a set of  independent and identically distributed (i.i.d.) random variables, every possible ordering of those variables is **equally likely**.\r<BR>\n" );
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document.write( "1. Let the first five observations be .<BR>\n" );
document.write( "2. Let the next observation be .<BR>\n" );
document.write( "3. We now have a total of **6 observations** in our set.\r<BR>\n" );
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document.write( "In a set of 6 independent observations from the same continuous distribution, any one of them is just as likely to be the \"minimum\" as any other.\r<BR>\n" );
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document.write( "### Calculation\r<BR>\n" );
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document.write( "Since there are 6 observations and each has an equal chance of being the smallest value:\r<BR>\n" );
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document.write( "* The probability that  is the smallest is .<BR>\n" );
document.write( "* The probability that  is the smallest is .<BR>\n" );
document.write( "* ...and so on.\r<BR>\n" );
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document.write( "The probability that the **next observation ()** is the smallest (less than all of the first five) is:\r<BR>\n" );
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document.write( "**The probability is 1/6 (or approximately 0.1667).**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Key Intuition\r<BR>\n" );
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document.write( "It does not matter what the actual distribution is (Normal, Uniform, Exponential, etc.). As long as the observations are independent and come from the same continuous distribution, the \"rank\" of the sixth observation is distributed uniformly among the possibilities .\r<BR>\n" );
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document.write( "Would you like to see how this probability changes if we were looking for the probability that the sixth observation is the *median* instead of the *minimum*? </SPAN>\n" );
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