A certain plant species has either red, pink, or white...

A certain plant species has either red, pink, or white flowers, depending on its genotype. If you cross a pink plant with any other plant, the probability distribution of the offspring is prescribed by the transition matrix $T=\left(\begin{array}{ccc}5 & .25 & 0 \\ .5 & .5 & .5 \\ 0 & .25 & .5\end{array}\right)$. On average, if you continue crossing with only pink plants, what percentage of the three types of flowers would you expect to see in your garden?

Key Concepts

Stationary Distribution

The stationary distribution is a probability distribution over the states of a Markov chain that remains unchanged by further transitions. It is found by solving the system of linear equations where the distribution is equal to itself multiplied by the transition matrix. This distribution represents the long-run behavior of the system, regardless of the initial state.

Steady-State Analysis

Steady-state analysis involves determining the stationary distribution of a system modeled by a Markov chain. It is used to predict the long-term proportions of each state when the process is run for many iterations. The analysis is crucial in understanding how the system evolves over time and what the expected equilibrium outcome is.

Markov Chain

A Markov chain is a stochastic process that transitions from one state to another within a finite or countable state space, where the probability of each state only depends on the previous state. This memoryless property simplifies analysis, especially when determining long?term behavior or steady-state probabilities of the system.

Transition Matrix

The transition matrix is a square matrix that represents the probabilities of moving from one state to another in a Markov chain. Each entry in the matrix corresponds to the probability of transitioning from a given initial state to a particular final state, and the rows of the matrix sum to one, reflecting the total probability distribution over the next state.

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