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Lesson: Stochastic Matrix

Worksheet • 2 Questions

Q1:

On any given day, an animal eats either cheese, grapes, or lettuce. The food the animal eats is influenced by the food it ate the previous day:

  1. If it eats cheese today, tomorrow it will eat either lettuce or grapes, with equal probability, but not cheese.
  2. If it eats grapes today, tomorrow it will eat grapes with a probability of , cheese with a probability of , or lettuce with a probability of .
  3. If it eats lettuce today, tomorrow it will eat grapes with a probability of or cheese with a probability of , but not lettuce.

Represent these probabilities in the matrix , where the columns, in order, represent the events of eating cheese, grapes, or lettuce tomorrow; and the rows, in order, represent the events of eating cheese, grapes, or lettuce today.

  • A
  • B
  • C
  • D
  • E

Letting , what is the interpretation of the entry ?

  • Athe probability that the animal eats grapes given that two days earlier the animal ate cheese
  • Bthe expected number of grapes the animal eats
  • Cthe probability that the animal eats cheese given that two days earlier the animal ate grapes
  • Dthe expected amount of cheese the animal eats
  • Ethe probability that the animal does not eat lettuce

Q2:

What does it mean for a matrix 𝐴 = ( π‘Ž ) 𝑖 𝑗 to be stochastic?

  • A 𝐴 has nonnegative entries and ο„š π‘Ž = 1 𝑗 𝑖 𝑗 .
  • B 𝐴 has negative entries and ο„š π‘Ž > 1 𝑗 𝑖 𝑗 .
  • C ο„š π‘Ž = 1 𝑗 𝑖 𝑗
  • D 𝐴 has nonnegative entries and ο„š π‘Ž β‰₯ 1 𝑗 𝑖 𝑗 .
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