3. Chapter 11, Exercise 4 (pg 414)
(a) Calculate the answer by figuring out all cases where grandsons have the desired outcome.
(b) Calculate the answer again, but this time by using matrix multiplication.
4. Chapter 11, Exercise 11 (pg 414).
(a) Calculate the answer by figuring out all cases where grandsons have the desired outcome.
(b) Calculate the answer again, but this time by using matrix multiplication.
(c) Use Wolframalpha (or similar) to find P
10 and P
100
.
(d) Use the matrix powers to estimate the probability that:
i. a distant descendant of a professional will be a professional?
ii. a distant descendant of a skilled laborer will be a professional?
iii. a distant descendant of an unskilled laborer will be a professional?
The Stationary Distribution
See the free textbook “Introduction to Probability” by Grinstead and Snell.
1 Let P denote the transition probability matrix used in the solution to 11.1.4(b) on the previous
homework.
(a) Compute P
10. What does this tell you about the probability that a 10th generation
descendant of a Darmouth alum will go to Dartmouth? Repeat for a 50th generation
descendant.
(b) The rows of P
50 are identical, or nearly so. Use this to determine the stationary distribution w (sometimes called π).
(c) Verify that indeed w P = w.
2 Repeat the previous 3 parts, but for 11.1.11.
Copyright ©2021 Ravi Montenegro
1
Matrix Operations with WolframAlpha
Inputting a matrix: Curly brackets should enclose the matrix and each row. For example,
{{1, 2, 3}, {0, 1, 2}, {−1, 0, 1}}
is the matrix
1 2 3
0 1 2
−1 0 1
Column vectors: A column vector is entered like
{{1}, {2}, {3}}
to mean
1
2
3
Matrix multiplication: Write A . B for the matrix product AB. For example:
{{1, 2, 3}, {0, 1, 2}, {−1, 0, 1}} . {{1}, {2}, {3}}
means
1 2 3
0 1 2
−1 0 1
1
2
3
Matrix Powers: The exponentiation symbol sometimes behaves strangely when applied to
matrices. It is better to use the MatrixPower command, such as:
M atrixP ower[{{1, 2, 3}, {0, 1, 2}, {−1, 0, 1}}, 5]
means
1 2 3
0 1 2
−1 0 1