The $p$ and $q$ are the wrong way around in the equation for $x_0$ given in (ii). This should be

\[

x_0 = 1 + p x_1+ q x_{-1}.

\]

\[

x_0 = 1 + p x_1+ q x_{-1}.

\]

Course information, a blog, discussion and resources for a course of 12 lectures on Markov Chains to second year mathematicians at Cambridge in autumn 2012.

The $p$ and $q$ are the wrong way around in the equation for $x_0$ given in (ii). This should be

\[

x_0 = 1 + p x_1+ q x_{-1}.

\]

\[

x_0 = 1 + p x_1+ q x_{-1}.

\]

If you are looking at past Markov chains tripos questions you might like to know that it has recently come to light that there is a mistake in 2012/3/9H.

The answer asked for, $1/(2k)$, should actually be $1/(2k+1)$.

The transition matrix $P$ is 1/4 of

2 2 0 0 0 0 0 0

1 2 1 0 0 0 0 0

0 1 2 1 0 0 0 0

0 0 1 2 1 0 0 0

0 0 0 1 2 0 0 0

0 0 0 0 1 2 1 0

0 0 0 0 0 1 2 1

0 0 0 0 0 0 1 3

(here illustrated for $k=7$) and so detailed balance gives the invariant distribution as proportional to (1,2,2,2,2,2,2,2).

Unfortunately, the examiner had put 0 0 0 0 0 0 2 2 in the bottom row and so calculated the invariant distribution as proportional to (1,2,2,2,2,2,2,1). But if the runner starts with $k$ shoes at the front door, the probability that she has $k$ shoes there after the next two runs is 3/4, not 1/2.

This question was intended to be similar to Example Sheet 2, #11 (the professor carrying umbrellas between home and office.)

The answer asked for, $1/(2k)$, should actually be $1/(2k+1)$.

The transition matrix $P$ is 1/4 of

2 2 0 0 0 0 0 0

1 2 1 0 0 0 0 0

0 1 2 1 0 0 0 0

0 0 1 2 1 0 0 0

0 0 0 1 2 0 0 0

0 0 0 0 1 2 1 0

0 0 0 0 0 1 2 1

0 0 0 0 0 0 1 3

(here illustrated for $k=7$) and so detailed balance gives the invariant distribution as proportional to (1,2,2,2,2,2,2,2).

Unfortunately, the examiner had put 0 0 0 0 0 0 2 2 in the bottom row and so calculated the invariant distribution as proportional to (1,2,2,2,2,2,2,1). But if the runner starts with $k$ shoes at the front door, the probability that she has $k$ shoes there after the next two runs is 3/4, not 1/2.

This question was intended to be similar to Example Sheet 2, #11 (the professor carrying umbrellas between home and office.)

This year's course finished on 13 November 2011. I handed out the standard faculty feedback forms in the lecture on 8 Novembers. I obtained 63 forms back.

You can also give feedback using my equivalent on-line form. You may find it easier to tick the radio buttons and type into this form than to handwrite onto the paper form - and you have space to write interesting comments. 8 of these have been received so far.

This will be sent to my email anonymously. After reading the responses, I will forward printed copies to the Faculty Office.

You can also give feedback using my equivalent on-line form. You may find it easier to tick the radio buttons and type into this form than to handwrite onto the paper form - and you have space to write interesting comments. 8 of these have been received so far.

This will be sent to my email anonymously. After reading the responses, I will forward printed copies to the Faculty Office.

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