Suppose a particle starts at position 5 on a number line and at each period the particle moves one position to the right with probability $p$ and, if the particle is above position $0$, moves one position to left with probability $1-p$.
Let $X_n(p)$ be the position of the particle at time $n$ for the given value of $p$. Use coupling to show that $X_n(b)st \leq X_n(a)$ for any $n$ if $b \leq a$.
I understand the question and I think I even know what to do. I believe it will be helpful to construct a coupling to introduce a family ${U_n}$ for independent $U(0,1)$ random variables and define coupling ${\hat{X}_n}$ recursively.
Is this correct and if it is, can someone please show me how to do it? I have a Statistic exam coming up soon so I am trying to do these tough problems as practice. By the way this is a question taken from the textbook A second Course in Probability by Sheldon Ross and its from chapter 2 question 2.
Thanks in advance
What is the question? For every $c$, decide that $X_n(c)$ moves to the right at time $n$ if and only if $U_n\leqslant c$. For every $a\leqslant b$, if $X_0(a)\leqslant X_0(b)$ then $X_n(a)\leqslant X_n(b)$ for every $n$ because each time $X_n(a)$ moves to the right, $U_n\leqslant a$ hence $U_n\leqslant b$, thus $X_n(b)$ moves to the right as well.