Relations of $\chi^2$ random variable and normal random variable and answering certain problems

Question

Correct me if I'm wrong. If $X$ is a random variable and if your trying to find the probability of $$ p(X^2 > 1) $$ you would use the chi-square ($\chi^2$) table so if you have a problem asking something along the lines of $$ p(X^3 > X) $$ couldn't you divide an $X$ getting $$ p(X^2>1) $$ and solve this using the Chi Square?

Answer 1

First of all, please label your random variables with capital letters. Lower case letters are for numbers, not variables.

Regarding your question, to make it more precise, we cannot divide by $X=0$, if it occurs. However, you are correct because by conditioning, \begin{align} P(X^3>X)&=\underbrace{P(X=0)}_{=0}\underbrace{P(X^3>X\,|X=0)}_{=0}+\underbrace{P(X\neq0)}_{=1}P(X^3>X\,|X\neq0) \\ &=P(X^2>1\,|X\neq 0) \\ &=P(X^2>1). \end{align}