Definition:
We say that $\{X(t)\}$ is a strict-sense stationary process if the joint distribution of any set of samples does not depend on the placement of the time origin, or in other words:
$$ \forall \tau \quad \forall k \quad \forall t_1,\dots,t_k \quad F_{X(t_1),X(t_2),\dots,X(t_k)}(x_1,x_2,\dots,x_k)=F_{X(t_1+\tau),X(t_2+\tau),\dots,X(t_k+\tau)}(x_1,x_2,\dots,x_k) $$
Question:
Assume that $A\sim\mathrm{Uniform}(2,9)$ is a random variable. Prove that the $\{X(t)=tA\}$ is not a strict-sense stationary process.
My try:
To prove that it's not strict-sense stationary (SSS), I tried to fix the variables. I assumed $\tau=1$, $k=2$, $t_1=1$, and $t_2=2$. If $\{X(t)\}$ is really SSS, then we should have:
$$ F_{X(1),X(2)}(x_1,x_2) = F_{X(2),X(3)}(x_1,x_2) \implies \mathbb{P}(X(1)\le x_1, X(2)\le x_2)=\mathbb{P}(X(2)\le x_1,X(3)\le x_2)\implies \mathbb{P}(A \le x_1, 2A \le x_2)=\mathbb{P}(2A\le x_1, 3A \le x_2) $$
Now, if I show that the above equation does not hold at least for some particular values of $x_1$ and $x_2$, we can conclude that $\{X(t)\}$ is not strict-sense stationary. However, this approach has some problems:
(I) I am not sure if this is the approach to prove that some given stochastic process is not strict-sense stationary. I mean, if we are given a process and asked if it is SSS, is this the best way to answer the question?!
(II) I cannot find the $x_1$ and $x_2$ that I'm looking for! It seems like I'm stuck.
Taking $k=1$ in the definition you see that the distribution of $X_t$ has to be the same for all $t$. Can you show that $X_1=A$ and $X_2=2A$ do not have the same distribution?