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.
It is simple to do both questions using the same approach as I explained here: http://www.mathhelpforum.com/math-he...ge-159842.html
Note that for the first question, note that the support of Y is and that the cdf of Y is since c < 0 (all working is left for you to do).