For a, b in (−∞, ∞), a < b, show that Y = a+(b−a)X has a uniform distribution over [a, b].

Question

Let X be uniformly distributed over the interval [0, 1] For a, b in (−∞, ∞), a < b, show that Y = a+(b−a)X has a uniform distribution over [a, b].

How would you approach this problem?

Uniform Distribution - Show an Expression is Uniform on (a, b)

This link has an approach to this question but couldn't understand the steps for the moment generating function?

Answer 1

I don't understand why one would use MGF for this . You get it from definition: $P(Y<c)=P(X<\frac {c-a} {b-a}) =\frac {c-a} {b-a}$ for $c$ between $a$ and $b$ and this is the definition of uniform distribution.