Proving Negative of Standard Normal is Standard Normal

Let X be standard normal random variable $N(1, 0)$ prove that $-X$ is also standard normal.

I think I am stuck on a technicality but here is my attempt:

Let $Y = -X$

P(Y $\leq$ u) = P($-X$ $\leq$ u) = P($X$ > -u) = 1 - P($X$ $\leq$ -u) = 1 - (1 - P($X$ $\leq$ u) = P($X$ $\leq$ u) which proves the result.

However I am not sure if my last step 1 - P($X$ $\leq$ -u) = 1 - (1 - P($X$ $\leq$ u) is justified. I would appreciate a hint.

Solution 1:

You need to know that the distribution is symmetric to prove your result.

You can show it is symmetric looking at the density $$\phi(-x)=\dfrac{1}{\sqrt{2\pi}}e^{-((-x)^2)}=\dfrac{1}{\sqrt{2\pi}}e^{-(x^2)}=\phi(x)$$