I have a variable $X\sim Exp(2)$ and i want a value $a$ with $0<a<1$ so the events that $X\in[0,1]$ and $X\in[a,2]$ are independant.
The only real progress I've managed to make is to find the probabilities of those intervals which were $1-e^{-2}$ for $x\in [0,1]$ and $-e^{-4}+e^{-2a}$ for $X\in [a,2]$, this was an attempt of using the definition of Independence for discrete random variables so the next stepwould be setting that equal to the probability of $X\in [a,1]$
It just doest feel right,am i even on the right track or should i try something different. Thanks for any help.
So we have $$P(0\leq X\leq 1)\cdot P(a\leq X\leq 2) = P(a\leq X\leq 1)$$
Since $$P(b\leq X\leq c) = \int_b^c 2e^{-2t}dt = -e^{-2t}\mid _b^c = e^{-2b}-e^{-2c}$$ we get
$$(1-e^{-2})(e^{-2a}-e^{-4}) = e^{-2a}-e^{-2}$$
Now you have to solve this equation...
$$e^{-2}-e^{-4}+e^{-6} = e^{-2a-2}$$
So $$1-e^{-2}+e^{-4} = e^{-2a} \implies a =-{1\over 2}\ln(1-e^{-2}+e^{-4})$$