Reference no: EM132309340

Question 1. A climate scientist makes the following observations on the weather in Ballarat in winter:

i There are never two consecutive days of nice weather.
ii Following a nice day, there are equal probabilities (i.e. 1) of having either a rainy day or a windy day.

iii Following a rainy day or a windy day, the probability of having the same weather the next day is 2, and the probability of having a nice day is 1.

a) What is the state space of this process?
b) Write down the Markov chain modelling the weather in Ballarat according to these ob-servations.
c) Show that this is a regular Markov chain.
d) Find the equilibrium (stationary) distribution of the Markov chain.
e) In the 3 months (90 days) of winter, how many days of nice weather can be expected?

Question 2. Let u(x, t) be a function of two variables which satisfies the following partial differential equation:

∂u/∂x + u/x = 3x

a) Solve to determine n. (Note that the PDE contains only u, derivatives of u, and functions of x.)

b) Find the solution which satisfies u(1, t) = sin²t.

c) Find any non-constant solution to the following partial differential equation:

∂²u/∂x∂y + 1/x∂u/∂y = 0

Question 3. Rewrite the third-order differential equation cos ty"'(t)+ √ty'(t) - e^ty(t) = 2t as a system of first-order differential equation.