Question

Show that the stationary probabilities for the Markov chain with transition probabilities πᵢⱼ are also stationary probabilities for the Markov chain with transition probabilities qᵢⱼ given by:

a) πᵢⱼ = qᵢⱼ

b) πᵢⱼ ≠ qᵢⱼ

c) πᵢⱼ > qᵢⱼ

d) πᵢⱼ < qᵢⱼ

Answer 1

The question seems to ask about showing the stationarity of probabilities in a Markov chain under different conditions, but the provided information is unclear and lacking context. A general approach to validate stationarity would be to demonstrate that the stationary distribution equation still holds with altered transition probabilities.

Step-by-step explanation:

The student's question appears to be asking how to show that certain stationary probabilities for a Markov chain remain stationary when the transition probabilities of the chain are altered. However, the information provided seems to be disjointed and out of context. The accurately posed question would require clear definitions of the transition matrices π_{i,j} and q_{i,j}, and how they relate to each other in the specific context of the Markov chain being analyzed. A general principle in Markov chains is that the stationary distribution π satisfies π = πP, where P is the transition matrix.

To determine if a given stationary distribution remains valid when transition probabilities change, we would need to verify if the equation still holds with the updated transition probabilities. Unfortunately, without specific values or a clear context, we cannot accurately demonstrate this. Moreover, information such as probability distributions directed along axes, retirement ages, hypotheses testing, particle decay, and reversible processes seem unrelated to the main question about Markov chains.