Question

If the random process is such that, "Future behavior of the process depends only on the present state and not on the past", then it is a:

• A. Poisson Process
• B. Binomial Process
• C. Markov Process
• D. Stationary Process

Hint:

The Markov property is a key principle in probability and stochastic processes, used in Markov Chains, Hidden Markov Models, and many real-world applications like speech recognition and finance.

Answer 1

The Correct Option is C

Step 1: Understanding the given condition The given definition describes a Markov Process, where the future state of the process depends only on the current state and is independent of past states.
Step 2: Defining Markov Property A stochastic process \( X_t \) is said to have the Markov property if: \[ P(X_{t+1} | X_t, X_{t-1}, ..., X_0) = P(X_{t+1} | X_t) \] This property ensures that only the present state influences future states, not the past states.
Step 3: Comparing with other processes
- Poisson Process: A counting process where events occur at a constant rate.
- Binomial Process: A process involving discrete trials with fixed probability.
- Stationary Process: A stochastic process where statistical properties do not change over time. Since the given property explicitly defines a Markov Process, it is the correct answer.