Modeling the probability of a uniform point on [a,b]I was given the following exercise : Explain how to define a probability measure on R that corresponds to the intuitive notion of an uniform point on [a,b].And this is the solution:Let f : R → R be defined as f ( x ) := ( b − a ) x + a. This function is an affine bijection between [0,1] and [a,b]. Let λ be the Lebesgue measure on [0,1]. We can define a probability measure µ on R by setting µ ( B ) := λ ( f − 1 ( B &gt; ∩ [ a , b ] ) ) for any Borel measurable subset B of R n . The corresponding c.d.f. is given by F ( x ) = 0 if x &gt; b, F ( x ) = 1 if x &gt; b and F ( x ) = f − 1 ( x ) = x − a b − a if x ∈ [ a , b ].I do not understand the question the exercise is asking. What is "the intuitive notion of an uniform point on [a,b]" ? Are they looking for a probability where all points in [a,b] have the same probability ?

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Modeling the probability of a uniform point on [a,b]I was given the following exercise : Explain how to define a probability measure on R that corresponds to the intuitive notion of an uniform point on [a,b].And this is the solution:Let   f  :      R    →      R   be defined as   f  (  x  )  :=  (  b  −  a  )  x  +  a. This function is an affine bijection between [0,1] and [a,b]. Let   λ be the Lebesgue measure on [0,1]. We can define a probability measure       µ   on R by setting       µ    (  B  )  :=      λ    (      f          −      1        (  B  &amp;gt;  ∩  [  a  ,  b  ]  )  ) for any Borel measurable subset B of             R        n  . The corresponding c.d.f. is given by   F  (  x  )  =  0 if   x  &amp;gt;  b,   F  (  x  )  =  1 if   x  &amp;gt;  b and   F  (  x  )  =      f          −      1        (  x  )  =            x      −      a              b      −      a       if   x  ∈  [  a  ,  b  ].I do not understand the question the exercise is asking. What is &quot;the intuitive notion of an uniform point on [a,b]&quot; ? Are they looking for a probability where all points in [a,b] have the same probability ?

Martha Dickson · Accepted Answer

Step 1Uniform means that the probability of a Borel set should be proportional to it&#039;s Lebesgue measure. Therefore we should define   μ  (  E  )  =      1          b      −      a        m  (  E  ), where m is Lebesgue measure.Step 2Their solution is a strange way to arrive at the same   μ.

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