Question

1. Suppose that we catch either salmon (state 1) or sea bass (state 2) according to a Markov Model where the transition matrix is given by 0.8 0.2 A= 0.4 0.6 Suppose that we compare the length, x, of each fish that we catch to 15cm, and that for all times P(x> 15| salmon) 0.2 P(x > 15 sea bass) 0.7 (a) Suppose we caught a salmon at time 1. What is the probability that we catch a sea bass at time 2? (b) Suppose we caught a salmon at time 1. What is the probability the next fish we catch (at time 2) has length less than or equal to 15cm?

Answer 1

a) $X_1=1$ corresponds to catching a salmon

We want to find $P(X_2=2|X_1=1)$

This is given by the matrix element

.412 = 0.2

b) We want to find $P(L_2\le 15|X_1=1)$

Which is $1-P(L_2>15|X_1=1)$

But $P(X_2=2|X_1=1)P(L_2>15)+P(X_2=1|X_1=1)P(L_2>15)$

Which is $P(X_2=2|X_1=1)=0.2,\,P(X_2=1|X_1=1)=0.8$

And $P(L_2>15|X_2=2)=0.7,\,L(L_2>15|X_2=1)=0.2$

So the probability is

PUs > 15X, = 1) 0.2 . 0.7+0.8. 0.2 0.3

Required probability is $1-P(L_2>15|X_1=1)=1-0.3={\color{Red} 0.7}$ is the probability that the next fish we catch will have length less than or equal to 15 cm given that the first fish was a salmon