Consider a Galton desk with 4 rows and 10 pins: We label the pins from 0 to 9, as shown in the figure. The arrows show some examples of how the ball bounces from one row to the next. (When the ball reaches a pin at the bottom row, it stays there and doesn't go anywhere.) At time 0 the ball begins at pin O. So at time 0, the probability that the ball is at 0 is equal to 1. We write this as: p (0) = 1; p₂(0)= 0; p₂(0) = 0..p₂(0) = 0. We can write this in vector form as: p(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]ª Let p(1) be the probability vector after 1 bounce. We have p(1) = [0, 0.5, 0.5, 0, 0, 0, 0, 0, 0, 0]¹ Let p() be the probability vector after j bounces. 1. Give the matrix T such that 2. Compute pl(2), p(3), p(4), p(5). pj + 1) = Tp(j)
Consider a Galton desk with 4 rows and 10 pins: We label the pins from 0 to 9, as shown in the figure. The arrows show some examples of how the ball bounces from one row to the next. (When the ball reaches a pin at the bottom row, it stays there and doesn't go anywhere.) At time 0 the ball begins at pin O. So at time 0, the probability that the ball is at 0 is equal to 1. We write this as: p (0) = 1; p₂(0)= 0; p₂(0) = 0..p₂(0) = 0. We can write this in vector form as: p(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]ª Let p(1) be the probability vector after 1 bounce. We have p(1) = [0, 0.5, 0.5, 0, 0, 0, 0, 0, 0, 0]¹ Let p() be the probability vector after j bounces. 1. Give the matrix T such that 2. Compute pl(2), p(3), p(4), p(5). pj + 1) = Tp(j)
Transcribed Image Text:Consider a Galton desk with 4 rows and 10 pins:
3
6
We label the pins from 0 to 9, as shown in the figure. The arrows show some examples of how the ball
bounces from one row to the next. (When the ball reaches a pin at the bottom row, it stays there and
doesn't go anywhere.)
At time 0 the ball begins at pin O. So at time 0, the probability that the ball is at 0 is equal to 1. We write
this as:
p₂(0) = 1; p₂(0) = 0; p₂(0) = 0..p₂(0) = 0.
We can write this in vector form as:
p(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let p(1) be the probability vector after 1 bounce. We have
p(1) = [0, 0.5, 0.5, 0, 0, 0, 0, 0, 0, 0]
Let p(j) be the probability vector after j bounces.
1. Give the matrix T such that
2. Compute pl(2), p(3), p(4), p(5).
pj + 1) = Tp(j)
Transcribed Image Text:Stochastic matrices
A Galton desk is a device that is used to demonstrate binomial probabilities. The desk consists of a set of
pins arranged in a triangular array.
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![Consider a Galton desk with 4 rows and 10 pins:
3
6
We label the pins from 0 to 9, as shown in the figure. The arrows show some examples of how the ball
bounces from one row to the next. (When the ball reaches a pin at the bottom row, it stays there and
doesn't go anywhere.)
At time 0 the ball begins at pin O. So at time 0, the probability that the ball is at 0 is equal to 1. We write
this as:
p₂(0) = 1; p₂(0) = 0; p₂(0) = 0..p₂(0) = 0.
We can write this in vector form as:
p(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let p(1) be the probability vector after 1 bounce. We have
p(1) = [0, 0.5, 0.5, 0, 0, 0, 0, 0, 0, 0]
Let p(j) be the probability vector after j bounces.
1. Give the matrix T such that
2. Compute pl(2), p(3), p(4), p(5).
pj + 1) = Tp(j)](https://content.bartleby.com/qna-images/question/eca4f090-b0ea-4e1b-8c25-7af952346ca8/6bd36a18-6002-4084-aed8-7b440de418ad/pq0yri8_processed.png)
