probability - Relation between the distribution functions of random variables $Y$ and $-Y$

I'm having trouble understanding a certain property of CDFs for negative random variables.

Let $Y$ be an exponential random variable and let $f_y, F_Y$ denote the PDF and CDF respectively.

My book claims that

$$f_{-Y}(y) = f_{Y}(-y)$$

I realized that I'm stuck on two parts.

1. I'm having trouble understanding firstly, the relationship between $-Y$ and $Y$.




2. I can't visualize the CDFs of $F_{-Y}$ and $F_{Y}$.

Any help would be appreciated. Thanks.

Answer

Let $Y$ have exponential distribution. It looks as if we are defining a new random variable $-Y$. We want the cumulative distribution function of $-Y$. The interesting part of the distribution function $F_{-Y}(w)$ is when $w$ is negative.

We have

$$F_{-Y}(w)=\Pr(-Y\le w)=\Pr(Y\ge -w)=1-F_Y(-w).\tag{1}$$

Note that this is different from what in OP is described as the book's claim.

Now differentiate to find $f_{-Y}(w)$. The differentiation introduces two cancelling minus signs, and from (1) we get $f_{-Y}(w)=f_Y(-w)$. Perhaps the book mistakenly used $F$ instead of $f$.