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Worksheet: Stochastic Matrix

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 1 1 0 , cheese with a probability of 4 1 0 , or lettuce with a probability of 5 1 0 .
  3. If it eats lettuce today, tomorrow it will eat grapes with a probability of 4 1 0 or cheese with a probability of 6 1 0 , but not lettuce.

Represent these probabilities in the matrix 𝑃 , 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 𝑃 =  0 0 . 4 0 . 6 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0 
  • B 𝑃 =  0 0 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0 
  • C 𝑃 =  0 0 . 4 0 . 6 0 . 4 0 . 1 0 . 4 0 . 6 0 . 5 0 
  • D 𝑃 =  0 0 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0 
  • E 𝑃 =  1 0 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 5 1 

Letting 𝑄 = 𝑃  , what is the interpretation of the entry 𝑄  οŽ•  ?

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

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 β‰₯ 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 = 1 𝑗 𝑖 𝑗 .