Showing a random Variable has a continuous uniform distribution

[LBJking123]

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.

 

[mathman]

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).

 

[LBJking123]

Ohh yeah that makes a ton of sense. Idk what I was thinking. Thanks!