given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection

Question

There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is:

$P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \land (0,0>t) $

given the variable Y=2X what is the exponential Cumulative distribution function $F_y(t)$? (they say that the distribution is a famous one)

I tried:

$P(2y \le t)=F_x(t) \to P(y \le t/2)=F_x(t/2)$

we create a new variable such that $T=t/2$ and so $F_y(T)=(1-e^{-T} , 0\le T) \land (0,0>T)$ but I got only that: $F_y(t/2)=(1-e^{-t/2} , 0\le t) \land (0,0>t)$ how it helps me to find $F_y(t)$? (this is not look like famous destribution...)

Answer 1

It seems you have many typos. I think the string of equalities you want to write goes as:

$$P(y \leq t ) = P(2x \leq t) = P( x \leq t/2) = F_x(t/2)$$

Then finish up from here.