<t>I think this problem can be set up recursively.For i), if we define the event of picking up a ball of the same color as the first as A, then... E[A] = (1/5)(1) + (4/5)(E[A]+1) => (1/5)E[A] = 1 => E[A] = 5For ii), we generalize the results from i)... E[A] = (p_i)(1) + (1-p_i)(E[A] + 1) => (p_i)E[A...

<t>MCarreira's method is probably the most straightforward if you can remember certain properties of the Markov matrix. Once you set up the transition matrix, it's an easy matter of finding the M block (the portion of the Markov matrix that includes only the transient states), and computing the inve...

<t>Define the stopping time to be the time the process hits -2 or 3. We can easily verfiy that the random walk and brownian motion are both martingales. Then, we can utilize the optional sampling theorem here.Since both the random walk and brownian motion satisfy the optional sampling theorem, if we...

<t>Since moving from one vertex to another vertex has equal probabilities, the long run probabilities of being in any position (A, B, C, D) should be equal -- namely, 1/4 for each vertex. The expected number of steps to reach state A starting from state A is then 1/(1/4) = 4. This expectations of re...