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Expected of each (Posted on 2014-05-15) Difficulty: 3 of 5
Randomly select n numbers from the continuous interval (0,1) and order them from least to greatest.

Find the expected values, in terms of n, of the smallest, second smallest, ..., second greatest, greatest.

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Solution computer simulation | Comment 1 of 3
 n               average value of:
     1st    2nd    3rd    4th    5th    6th   7th   8th    9th    10th
 2 0.3323 0.6626
 3 0.2486 0.5002 0.7495
 4 0.2027 0.4023 0.6017 0.8016
 5 0.1667 0.3358 0.5025 0.6667 0.8325
 6 0.1406 0.2843 0.4267 0.5695 0.7137 0.8557
 7 0.1264 0.2532 0.3779 0.5004 0.6241 0.7509 0.8761
 8 0.1106 0.2214 0.3337 0.4457 0.5567 0.6676 0.7778 0.8896
 9 0.0995 0.2007 0.3004 0.4019 0.5015 0.6023 0.7031 0.8020 0.9001
10 0.0911 0.1814 0.2722 0.3642 0.4536 0.5433 0.6353 0.7263 0.8165 0.9092
 
 and it appears the expected value of the mth smallest when n random numbers are chosen is m/(n+1). There were 10,000 trials at each value of n.
 
 DefDbl A-Z
 Dim crlf$

 Function mform$(x, t$)
   a$ = Format$(x, t$)
   If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
   mform$ = a$
 End Function
 
 Private Sub Form_Load()
  Text1.Text = ""
  For sz = 2 To 10
     ReDim v(sz), tot(sz)
     crlf$ = Chr(13) + Chr(10)
     Form1.Visible = True
     DoEvents
     For tr = 1 To 10000
       For i = 1 To sz
         v(i) = Rnd(1)
       Next
       Do
         done = 1
         For i = 1 To sz - 1
           If v(i) > v(i + 1) Then swap v(i), v(i + 1): done = 0
         Next
       Loop Until done = 1
       For i = 1 To sz
         tot(i) = tot(i) + v(i)
       Next
       If tr = 10000 Then
        Text1.Text = Text1.Text & mform(sz,"#0")
        For i = 1 To sz
         Text1.Text = Text1.Text & mform(tot(i) / tr, " 0.0000")
        Next
        Text1.Text = Text1.Text & crlf$
        DoEvents
       End If
     Next tr
  Next sz
 End Sub
 Sub swap(a, b)
   Dim h
   h = a: a = b: b = h
End Sub

  Posted by Charlie on 2014-05-15 16:35:46
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