Math for Quantitative Finance
# Expected Value

Here’s the problem we’ll be tackling:

An ant is at a vertex of a tetrahedron. Every second, it randomly chooses one of the other 3 vertices, and crawls to that vertex. What is the expected number of seconds until it has visited every vertex?

Let’s define some variables to help solve the original problem. Which do you think would be the most helpful?

**A:**\(X_i,\) the expected number of distinct vertices visited after \(i\) seconds**B:**\(X_i,\) the expected number of seconds until \(i\) distinct vertices have been visited

If \(X_i\) is the expected number of seconds until \(i\) distinct vertices have been visited. What is \(X_1?\)

Remember that the ant starts at a vertex (at time 0).

Using the same logic, you can determine the expected number of seconds between visiting the third vertex and final vertex.

Using this and all prior knowledge, what is \(X_4,\) the amount of total expected time in seconds from when the ant starts moving at 0 seconds?

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