Question: The Markov property does not imply that the past and the future are independent given *any* information concerning the present. Find an example of a homogeneous Markov chain $\left\{X\_n\right\}\_{n\ge0}$ with state space $E=\left\{1,2,3,4,5,6\right\}$ such that 

$P(X\_2=6\mid X\_1\in\{3,4\},X\_0=2)\neq P(X\_2=6\mid X\_1\in\{3,4\})$.

I don't understand the phrasing of this question. How should I approach this?

Could this work? [![enter image description here](https://i.stack.imgur.com/SLR2e.jpg)](https://i.stack.imgur.com/SLR2e.jpg)

$P(X\_2=6\mid X\_1\in\{3,4\}) = 1$

But $P(X\_2=6\mid X\_1\in\{3,4\},X\_0=2) =0$

Answer: 
**Hint $1$**: The fundamental theorem of calculus states that
$$\int\_a^b g(x)\,\mathrm dx = G(b) - G(a)$$
for a primitive $G$ of $g$. Use it to find $f'(x)$. What can you conclude now?

**Hint $2$**: To find the asymptotes, visualize the graph of the integrand function. This should give you an idea for $x \to -\infty$. For the other side, it boils down to the improper integral $\lim\limits\_{x \to +\infty} f(x)$. Does it converge?
Apply Newton-Leibnitz theorem. Then note that max or min is achieved at $0$ for derivative . Now differentiate the derivative to get second derivative by product rule. See for which of the root of first derivative the value of second derivative is positive(because thats the condition for minima. Hope you know why)