Proses Stokastik dan Rantai Markov

Summary

This video provides a comprehensive introduction to stochastic processes, a key topic in probability and statistics. It covers types of stochastic processes based on time and state space, with a focus on Markov Chains. The script explains the memoryless property of Markov processes, where future states depend only on the present, not the past. Through real-world examples such as phone sales and bacteria growth, the video explores both discrete and continuous time-state processes. Additionally, it delves into transition probabilities, key to understanding how states evolve over time in these processes, making the content accessible for exam preparation.

Takeaways

• 😀 Stochastic processes are random variables dependent on time, represented by Xt(t) or XT.
• 😀 Two key parameters in stochastic processes: time (discrete or continuous) and the state space (also discrete or continuous).
• 😀 The state value at a given time is referred to as the 'state' or 'step' of the process.
• 😀 Stochastic processes can be classified based on the type of time and state space: discrete time & discrete state, discrete time & continuous state, continuous time & discrete state, and continuous time & continuous state.
• 😀 Example of a discrete-time and discrete-state process: sales of phones in a store per day.
• 😀 Example of a discrete-time and continuous-state process: rainfall volume per day, which can be measured in decimals.
• 😀 Example of a continuous-time and discrete-state process: number of calls received at a call center during an interval.
• 😀 Example of a continuous-time and continuous-state process: temperature readings in a specific area over time.
• 😀 Markov chains are a special type of stochastic process where the future state depends only on the present state and not on the past states.
• 😀 Markov chains can be homogeneous, meaning the transition probabilities between states remain constant over time.

Q & A

What is a stochastic process?

A stochastic process is a random variable that depends on time. It is denoted as X(t) or X_t, representing the values of the process at different times.

What are the two main components of a stochastic process?

The two main components of a stochastic process are the time variable (which can be discrete or continuous) and the state space or outcome variable (which can also be discrete or continuous).

What does the term 'state' refer to in a stochastic process?

The term 'state' refers to the value or condition of the process at a specific point in time. These states can be either discrete or continuous, depending on the type of process.

Can you explain the four types of stochastic processes?

The four types of stochastic processes are: 1) Discrete time, discrete state space, 2) Discrete time, continuous state space, 3) Continuous time, discrete state space, 4) Continuous time, continuous state space.

What is an example of a stochastic process with discrete time and discrete state space?

An example is the number of mobile phones sold per day in a store. The time is discrete (each day) and the number of phones sold is also discrete (a whole number).

What is a Markov chain?

A Markov chain is a specific type of stochastic process that satisfies the Markov property, where the future state depends only on the current state and not on the sequence of events that preceded it.

What does the Markov property state?

The Markov property states that the probability of future events depends only on the current state and not on past states. This means that past events do not influence the future state.

What is a Markov chain with discrete time?

A Markov chain with discrete time is a process where the state space is discrete and the transitions between states happen at distinct time intervals. The process only depends on the current state and time.

What is the meaning of transition probability in a Markov chain?

Transition probability in a Markov chain refers to the likelihood of moving from one state (i) to another state (j) in a given time step. It is denoted as P(i -> j) and can be represented as a matrix of probabilities.

What is the difference between homogeneous and non-homogeneous Markov chains?

In a homogeneous Markov chain, the transition probabilities are constant over time, meaning they do not change as time progresses. In a non-homogeneous Markov chain, transition probabilities may change over time.