Is $X$ and $X-0$ the same random variable?

We have a random variable $X(t)$. We define $X(0)=0$. I am aware of the $Y$~$N(0,1), Z=-Y$ example, that is, even though 2 random variables have the same distribution and defined on the same probability space it doesn't mean that they're equal.

However what about this, We know $\forall_{t_1>t_2>0}, \left(X(t_1)-X(t_2)\right)$~$N(0,t_1-t_2)$. But then $\left(X(t)-X(0)\right)$~$N(0,t)$. But, $X(t)-X(0)=X(t)-0=X(t)$. While I can always just substitute $X(t)$ with $X(t)-X(0)$ as needed in my calculations, I would like to be precise. Is $X(t)-X(0)$ and $X(t)$ the same random variable?

Solution 1:

I’m assuming $X(t)$ is supposed to be a family of RVs, else the whole thing would not make sense. Generally a random variable is a function from the probability space to $\mathbb R$. So your question turns into:

Is $X_t - 0$ the same function as $X_t$? As both functions have the same domains this is true if $X_t(\omega)- 0 = X_t(\omega)$ for all $\omega\in\Omega$, which is clearly true.

So yes, $X-0$ is the same RV as $X$.