I was trying to solve an exercise about derived random variables, but I am stuck on deriving the boundaries to calculate expectation and variance.
The random variable $X$ is exponentially distributed with parameter $\lambda= 1/2$. Furthermore, define $Y= 1−2X$. Calculate $E(Y)$ and $\operatorname{Var}(Y)$.
This is what I came up with for now.
It would be much appreciated if someone could help.
Hint
$\Bbb{E} (a + bX)$ and $\text{Var}(a+bX)$ can be found in terms of $\Bbb{E}(X)$ and $\text{Var}(X)$ respectfully. This comes from the linearity of expectation.
Deriving the first formula, we have $$\Bbb{E}(a+bX) = \int_{- \infty}^\infty f(x)(a+bx) \, dx $$
$$\Bbb{E}(a+bX) = \underbrace{a \int_{- \infty}^\infty f(x)\, dx}_{= a} \,+ \underbrace{ b \int_{- \infty}^\infty x f(x) \, dx \, }_{=b \Bbb{E}(X)}$$
$$\Bbb{E}(a+bX) = a + b\Bbb{E}(X)$$
You may now use $a=1$ and $b=-2$ to solve for the mean of $Y$
A similar process can be used to derive the variance formula (and it is perhaps easier now that you can apply linearity of expectation)