Question

Let Y=2X-1. Find the distribution of Y if X is uniform on (-2,4).

Answer 1

First of all note that x~Uniform(-2,4) implies -2<=X<=4 and hence -4<=2X<=8 and we get -4-1<=2X-1<=8-1 so that

-5<=Y<=7.

Then P(Y<=y) =P(2X-1<=y)=P(X<=(y+1)/2).

Now P(X<=x)=0 if x<-2

= if -2<=x<4

=1 if x>=4

Then P(X<=(y+1)/2)=0 if (y+1)/2<-2

= ((y+1)/2+2)/6 if -2<=(y+1)/2<4

=1 if (y+1)/2>=4

That is   P(Y<=y)= P(X<=(y+1)/2)=0 if y<-5

= ((y+1)/2+2)/6 if -5<=y<7

=1 if y>=7

Then f(y)==1/12 if -5<=y<7

=0 otherwise

Hence Y has Uniform(-5,7) distribution.