Desperate! Involves matrices

Question

 

 

Question Two

 

Answer 1

(b)

For the year 2013 ,  62% of commuters travelled by car

That means 38% of commuters are using car

Initial probability matrix 

Given that the 8% of commuters who travel by car one year switch to bus  

Given that the 12% of commuters who travel by bus one year switch to car  

To find the probability Pn after “n” years we  multiply the initial matrix by the transition matrix raised to the power of “n”

 Pn = P x Tn

we have already calculated transition matrix 

So after one year (means in 2014)[P1]

In 2014 ,  61.6% of commuters travel by car 

 

 

After two year (means in 2015)[P2]

In 2015 ,  61.28% of commuters travel by car 

Answer 2

(a)

 A Transition matrix  (also termed probability matrix or Markov matrix) is a matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegitive real numbers representing a probability. It has found use in probability theory, statistics and linear algebra , as well as papulation genetics.

Here we have to write transition matrix corresponds to new public transport system

Given that the 8% of commuters who travel by car one year switch to bus  

That means 92% commuters remains in using car 8% switch to bus 

Given that the 12% of commuters who travel by bus one year switch to car  

That means 88% commuters remains in using bus , but  12% switch to car 

Answer 3

(c)

To find out the previous year percentage we use inverse of transition matrix concept

The probability Pn before “n” years  we  multiply the initial matrix by the transition matrix raised to the power of “-n”

 Pn = P x T(-n)

 

Before  one year  (means in 2012)[P(-1)]

Inverse of transition matrix 

In 2012 ,  62.5% of commuters travel by car 

 

Answer 4

(2)

Given transition matrix 

This matrix is regular since all entries are positive .Let T represents  transition matrix ,

and let V be the probability vector  .We want to find V such that 

 V×T=V

Multiplication of matrix

0.7a+0.3b = a     

0.3a = 0.3b

a = b  -----(1)

0.05a+0.95b = b   ------(2)

From (1) and (2)

a = 1 ,  b = 1

Proportion of a = 1

Proportion of b = 1