Worksheet: Stochastic Matrix

In this worksheet, we will practice using a matrix to represent probabilities.

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 110, cheese with a probability of 410, or lettuce with a probability of 510.
  3. If it eats lettuce today, tomorrow it will eat grapes with a probability of 410 or cheese with a probability of 610, 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 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0
  • B 𝑃 = 0 0 . 4 0 . 6 0 . 4 0 . 1 0 . 4 0 . 6 0 . 5 0
  • C 𝑃 = 0 0 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0
  • D 𝑃 = 1 0 . 5 0 . 5 0 . 4 0 . 1 0 . 5 0 . 6 0 . 5 1
  • E 𝑃 = 0 0 . 4 0 . 6 0 . 4 0 . 1 0 . 5 0 . 6 0 . 4 0

Letting 𝑄=𝑃, what is the interpretation of the entry 𝑄?

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

Q2:

What does it mean for a matrix 𝐴=(𝑎) to be stochastic?

  • A 𝐴 has nonnegative entries and 𝑎1.
  • B 𝐴 has nonnegative entries and 𝑎=1.
  • C 𝑎 = 1
  • D 𝐴 has negative entries and 𝑎>1.

Q3:

Under which of the following conditions is the 𝑛×𝑛 matrix 𝐴=(𝑎) a Markov matrix?

  • A 𝑎 0 for all 𝑖,𝑗 and 𝑎1
  • B 𝑎 > 0 for all 𝑖,𝑗 and 𝑎=1
  • C 𝑎 0 for all 𝑖,𝑗 and 𝑎=0
  • D 𝑎 0 for all 𝑖,𝑗 and 𝑎=1

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