Since we were not provided any solutions to our statistics exercises, I wanted to ask you guys for any corrections or errors I did on this exercise.
I will only upload a screenshot from my notes app. I hope that's ok.
The exercise was: Let $X$ be a random variable with density $fX : \mathbb{R} \to \mathbb{R}$ defined by fX(x) = (1 2 for x ∈ [2, 4] 0 else
a) Calculate the corresponding distribution function.
b) Determine the density of the random variable Y := 2X
Thank you :)
point b is correct even there is a faster way to solve it. Just remembering that with a monotone transformation function there is a suitable formula that is
$$f_Y(y)=f_X\left(g^{-1}(y)\right) \left|\frac{d}{dy}g^{-1}(y)\right|$$
Thus being $x=\frac{y}{2}$ and $x'=\frac{1}{2}$ you immediately get
$$f_Y(y)=\frac{1}{4}\cdot \mathbb{1}_{[4;8]}(y)$$
as you found.
a) is wrong because when $x\geq 4$ the CDF $F_X(x)=1$
(remember also that, when calculating an integral function, i.e. $\int_2^x f(t)dt$ you cannot differentiate in $dx$ but you have to use another letter; you can use the letter you prefer but not $x$)