Probability of a random variable and its counterpart raised to power of $n$

Question

I came across few situations where they stated that the probability of a random variable is equal to the probability of its square root (case1) or its square (case 2). My questions are:

1. How could the equalities be derived, based on what logic?

2. Can I make a general statement that the probability of a random variable is equal to its counterpart raised to the power of $n$ where $n$ is any real number, that is, $$\mathbb{P}(X \leq a) = \mathbb{P}(X^n \leq a^n)$$

Case 1:

Case 2:

Answer 1

Remember that anything appearing inside a $P(\cdot)$ is an event. More precisely, this means that if the random variables are defined on a probability space $\Omega$, then an event $A$ is a subset of $\Omega$ and $P(A)$ depends only on the the set of elements of $A$ (and not on how a description of $A$ appears).

Going back to your question, in both cases we have two descriptions of the same event. In the first case we have the event $$ A=\Biggl\{\omega\in\Omega\colon \frac{(X(\omega)-\mu)^2}{\sigma^2}\geq t^2\Biggr\}, $$ which is equal to the event $$ B=\Bigl\{\omega\in\Omega\colon |X(\omega)-\mu|\geq t\sigma\Bigr\}, $$ since if $\omega\in B$ then by squaring we see that $\omega\in A$, and conversely. The reasoning is similar in the second example as well.

Answer 2

For $x,y\ge 0$ and $a>0$, $x\le y$ iff $x^a\le y^a$. Therefore, for a nonnegative random variable $X$, $$ \{X\le y\} =\{X^a\le y^a\}. $$