help with simplifying Markov chains

Question

1) $p(X_1)p(X_2|X_1)p(X_3|X_1, X_2)p(X_4|X_1, X_2, X_3)$

2) $p(X_4)p(X_3|X_4)p(X_2|X_3, X_4)p(X_1|X_2, X_3, X_4)$

For the first one, I got : $p(X_1)\,p(X_2|X_1)\,p(X_3\mid X_2)\,p(X_4\mid X_3)$ and for the second one, I got : $p(X_4)\,p(X_3\mid X_4)\,p(X_2\mid X_3)\,p(X_1\mid X_2)$.

I am not sure if this is correct so it would be appreciated to see if it is.

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Answer 1

For the second one: $$ P(X_4)\cdot P(X_3|X_4)\cdot P(X_2|X_3, X_4)\cdot P(X_1|X_2, X_3, X_4) \\ = P(X_4) \cdot \frac{P(X_3,X_4)}{P(X_4)} \cdot \frac{P(X_2,X_3,X_4)}{P(X_3,X_4)} \cdot \frac{P(X_1,X_2,X_3,X_4)}{P(X_2,X_3,X_4)} \\ P(X_1,X_2,X_3,X_4) $$

The first one can be simplified in the same way.

You may also be right, I'm not really sure. If you think you are right, you could show me how you got your answer?