Difference equations
(a) Consider the inhomogeneous difference equation
(n + 1)^3yn+1 n^3yn = 3
Give the corresponding homogeneous difference equation and solve it!
Note: It was easier for me than I found the fraction , which occurred to me at the
n^3(n + 1)^ 3
Solution came when I wrote a double fraction!
(b) Now solve the inhomogeneous difference equation by writing the c from point (a) as function c (n) and inserting it into the inhomogeneous equation!
(c) Make sure that the consequence
yn = 2(3^n 1) + 3^nY0 the Diefference equations yn+1 = 3yn + 4 fullfils
Exercise 2:
Markov processes: A math lecturer likes to eat pudding for breakfast. Since he values a balanced diet, he alternates between chocolate pudding and vanilla pudding, with the following transition probabilities:
If he chooses chocolate on a given day, there is a probability 0.8 that he will choose chocolate again the next day.
However, once he has decided on vanilla, there is a probability of 0.7 that he will spoon up custard again the next day.
In the following we denote the probability distribution for each variety on the nth day with the vector p (n); the first component indicates the probability of chocolate on the nth day, the second vanilla – ie: p (5) = (0.9, 0.1) means: there is a 90% probability that he will have chocolate pudding for breakfast on the fifth day , with 10% probability vanilla.
Draw the Markov diagram and give the transition matrix.
(b) Assuming it starts with chocolate – i.e. p (0) = (1, 0): Find the probability distributions p (1), p (2) and p (10) by matrix-vector multiplication! . Please also carry out the first two calculations by hand, p (10) with a program of your choice – Wolfram Alpha can do it, but note that you multiply the vector from the left by the matrix!
Now calculate the same distributions if he started with vanilla (electronically enough!)
Calculate the probabilities even if he is undecided on day 0 and he threw a coin, but we do not know the outcome!
Now draw all p (n) obtained in this way in a coordinate system – by hand or with R, as you wish – and connect them! Think about what the connecting line means!
(c) Now calculate the two – left again! – Eigenvalues 1, 2 and the eigenvectors e1, e2 of the matrix (please by hand!). mark them in the diagram (b). Find an interpretation for both vectors!
(d) (Difficult – 2 points!) Assume that p (0) = (0.8, 0.2). Find p (1)! Now represent p (0) and p (1) as a linear combination of the eigenvectors:
p(0)=ae1 +be2, p(1)=ce1 +de2.
Compare the pairwise the coefficients and think about how to use this with
the linearity of the vector matrix multiplication is related!
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