Density function of a negative random variable

Question

Let $X$ be an independent random variable uniformly distributed in $(0, 1)$

Hence I have $f_X(x)=\mathbb{1}_{(0,1)}(x)$, but what can I say about $f_{-X}(x)$?

Sorry, maybe it's a trivial question, but I never worked with negative variables. Any help/hint?

Is $f_{-X}(x)=\mathbb{1}_{(0,1)}(-x)=f_X(-x)$?

Answer 1

Hint: We just introduced a new random variable

$$Y=-X,$$ ditributed in $(-1,0)$, so we can try to understand its cdf by using our understanding of $X$: $$F_Y(y)= P(Y \leq y) = P(X\geq -y) = 1- F_X(-y). $$

We can then differentiate to get the pdf of $Y$.