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Product Chain
Let \(Z_n\) be a collection of independent random variables with \(P(Z_n=1)=\frac12\) and \(P(Z_n=\frac12)=\frac12\) . Define \(X_0=1\) and \(X_{n+1}=Z_n X_n\).
- What is \(E( X_n | X_{n-1})\) ?
- What is \(E(X_n)\) ?
- What is \(\mathrm{Cov}(X_n,X_{n-1})\) ?
Basic Markov Chain I
In each of the graphs pictured, assume that each arrow leaving a vertex has an equal chance of being followed. Hence if there are thee arrows leaving a vertex then there is a 1/3 chance of each being followed.
- For each of the six pictures, find the Markov transition matrix.
- State if the Markov chain given by this matrix is irreducible.
- If the Matrix is irreducible, state if it is aperiodic.
Selling the Farm
Two competing companies are trying to buy up all the farms in a certain area to build houses. In each year 10% of farmers sell to company 1, 20% sell to company 2, and 70% keep farming. Neither company ever sells any of the farms that they own. Eventually all of the farms will be sold. Assuming that there are a large number of farms initially, what fraction do you expect will be owned by company 1 ?
[Durrett “Elementary Probability”, p 159 # 39]
Computers on the Blink
A university computer room has 30 terminals. Each day there is a 3% chance that a given terminal will break and a 72% chance that that a given broken terminal will be repaired. Assuming that the fates of the various terminals are independent, in the long run what is the distribution of the number of terminals that are broken ?
[Durrett “Elementary Probability” p. 155 # 24]
Conditioning and Polya’s urn
An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.
- Let \(B_n\) be the number of black balls in the urn just before the \(n\)th ball is drawn. (Thus \(B_1= 1\).) For \(n \geq 1\), find \(\mathbf{E} (B_{n+1} | B_{n}) \).
- For \(n \geq 1\), find \(\mathbf{E} (B_{n}) \). [Hint: Use induction based on the previous answer and the fact that \(\mathbf{E}(B_1) =1\)]
- For \(n \geq 1\), what is the expected proportion of black balls in the urn just before the \(n\)th ball is drawn ?
[From pitman p 408, #6]
Expectation of mixture distribution
Consider the following mixture distribution.
- Draw \(X \sim \mbox{Ber}(p=.3)\)
- If \(X=1\) then \(Y \sim \mbox{Geometric}(p_1)\)
- If \(X= 0\) then \(Y \sim \mbox{Bin}(n,p_2)\)
What is \(\mathbf{E}(Y)\) ?. (*) What is \(\mathbf{E}(Y | X )\) ?.
