Let X~Uniform[L,R]. Let Y=cX+d, where c<0. Prove that Y~Uniform[cL+d,cR+d]. (In particular if L=0 and R=1 and c=-1 and d=1, then X~Uniform[0,1] and also

Y=1-X~Uniform[0,1])

my attempt

Y=cX+d

X= (Y-d)/c since c<0

(Y-d)/c ~ Uniform[L,R] since c<0

isolating for Y gets you

L <= (Y-d)/c <= R since c<0

cL <= Y-d <= cR

cL+d <= Y <= cR+d

Y~Uniform[cL+d,cR+d]

Y~Uniform[(-1)0+1,(-1)1+1]

Y~Uniform[1, 0]

1 <= -X+1 <= 0

0 <= -X <= -1

0 <= X <= 1

X~Uniform[0,1]

Y=1-X~Uniform[0,1]

cX+d=1-X~Uniform[0,1]

X~Uniform[0,1] = 1-(cX+d)

X~Uniform[0,1] = 1-cX-d

X~Uniform[0,1] = 1+X-1

X~Uniform[0,1] = X

is all this correct?

also

another question

Let X~exponential(lambda). Let Y=cX where c>0. prove that Y~Exponential(lanbda/c)

i don't understand how to do this proof.