A system has three possible states, 0, 1 and 2. Every hour it makes a transition to a different state, which is determined by a coin flip. For example, from state 0, it makes a transition to state 1 or state 2 with probabilities 0.5 and 0.5. a. Find the transition probability matrix.
b. Find the three-step transition probability matrix.
c. Find the steady-state distribution of the Markov chain.

Step-by-step explanation:

From the given information;

the transition probability matrix (TPM) for state 0, 1, 2 is:

The three-step TPM  is done by a simple matrix multiplication which is computed as follows:

The steady-state distribution of the Markov Chain is  determined by first solving the system πp = π together with the normalizing condition

here;

π₁ + π₂ + π₃ = 1

So;

0.5π₂ + 0.5π₃ = π₁

⇒ π₂ + π₃ = 2π₁

0.5π₁ + 0.5π₃ = π₂

⇒ π₁ + π₃ = 2π₂

0.5π₁ + 0.5π₂ = π₃

⇒ π₁ + π₂ = 2π₃

Thus, π₁ + π₂ + π₃ = 1

This can be re-written as:

1 - π₁ = 2π₁

1 - π₂ = 2π₂

1 - π₃ = 2π₃

∴

1 = 3π₁

π₁ = 1/3

1 = 3π₂

π₂ = 1/3

1 = 3π₃

π₃ = 1/3

Hence, the steady-state probability distribution is π = [1/3, 1/3, 1/3]

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .