Rectangular distribution

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Notes

For XRect([a,b]) where [a,b] denotes the closed interval {xR | axb} we have the following:

  • probability density function f:[a,b]R0 by f:x1ba - this can of course be extended to R by making it zero outside of [a,b]R
  • cumulative density function F:[a,b][0,1]R by F:xxaba - this can also be extended by making it 0 before a and 1 after b

Properties are:

  • E[X]=12(a+b) - the average of a and b, unsurprisingly
  • Var(X)=112(ba)2
    • Giving S.D=123(ba), note that 233.4641
  • Mdm(X)=14(ba)

Note that the standard deviation (which has the same units as the mdm) is slightly larger than the mdm, the mdm is 86.60% (4 s.f) of the sd