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Showing a random Variable has a continuous uniform distribution

  1. Mar 26, 2013 #1
    f(x)=1, θ-1/2 ≤ x ≤ θ+1/2

    Given that Z=(b-a)(x-θ)+(1/2)(a+b) how would you show that Z has a continuous uniform distribution over the interval (a,b)?
    Any help would be much appreciated.
     
    Last edited: Mar 26, 2013
  2. jcsd
  3. Mar 27, 2013 #2

    mathman

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    Let y = x-θ. Then y is uniform over the interval (-1/2,1/2). Since Z is linear in y, it is also uniformly distributed.
    Put in the endpoints of y to get the endpoints for the Z interval. At y = -1/2, Z = a, while at y = 1/2, Z = b.

    If you want to do a little work, start with P(Z < z) and transform it into P(Y < y) or P(X < x).
     
  4. Mar 27, 2013 #3
    Ohh yeah that makes a ton of sense. Idk what I was thinking. Thanks!
     
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