MATH 410 Homework #10
(Due Monday, 27 April 2015)
Read: Dobrow, Chapter 7, §§3-4.
Problems:
1. Recall from class the “telegraph process” defined by Xt = (−1)Nt , where {Nt , t ≥ 0} is a
Poisson Process with rate λ.
(a) Find the transition matrix P (t).
Hint: First observe that Xt = 1 ⇔ Nt is even. Now write out the power series for ex
and e−x and add them term by term. How is this useful?
(b) Find lim P (t). Explain why the result makes intuitive sense.
t→∞
2. Three frogs are playing near the pond at the Center for the Arts on a sunny April day
in Middlebury. The frogs behave independently. A frog in the sun gets too hot after an
exponentially distributed time with mean 1 minute, and jumps into the pond. A frog in the
pond gets too cold after an exponentially distributed time with mean 12 minute and jumps
onto land. Let Xt be the number of frogs in the sun (i.e., on land) at time t.
(a) Draw the rate transition diagram for the process.
(b) Find the stationary distribution π.
(c) Check your result for π by noting that the 3 frogs represent independent two-state
flip-flop chains, and applying our analysis from class.
(d) Write the generator matrix Q for the process and verify that πQ = 0.
3. Recall from class the one-way Markov switch model, for which we derived the transition
matrix
0
0
P (t) =
1
1
1
0
1 − e−λt e−λt
(a) Use a rate transition diagram to explain the following form for the infinitesimal generator
0
Q =
0
1
1
0 0
λ −λ
(b) Verify directly that P 0 (t) = P (t)Q = QP (t). The first expression is an element-byelement derivative, the last two are matrix multiplies.
(c) Evaluate the matrix power series to show that
P (t) = eQt
where
eQt = I + tQ +
t2 2 t3 3
Q + Q + ···
2!
3!
(You should find that powers of Q settle into an obvious pattern.)