What is the mean of $Z^2$ (standard normal variable)?

Question

So I'm following a solution in my textbook on the $\chi^2$ distribution.

I am stuck on a step which involves finding the mean of $Z^2$, where $Z$ is a random variable from the standard normal distribution. I understand the mean of $Z$ is $0$, but the textbook claims the mean of $Z^2$ is $1$, and I can't work out why as it doesn't offer an explanation for this.

Can anyone explain this to me?

Answer 1

Note that $$ \text{var}Z = EZ^{2} - (EZ)^{2} = EZ^{2} = 1. $$