Probability distribution function symmetrical about origin

Question

If $X$ be random variable and it is continuously distributed with $f(-x)=f(x)$ then find $F(0)$ and show $$P(-a<X<a)=2F(a)-1$$

Answer 1

Just have a look at a symmetric distribution like the normal distribution. The random variable X is distributed as $X\sim \mathcal N(0,\sigma^2)$

In the picture below $F(a)$ represents the areas $1,2$ and $3$. Area $4$ can be calculated with the converse probability: $1-F(a)$. This area is equal to area $1$. If we substract area $1$ from F(a) we get the area from $-a$ to $a$.

$$P(-a<x<a)=F(a)-(1-F(a))=F(a)-1+F(a)=2F(a)-1$$