“Stochastic” refers to systems, processes, or phenomena that involve randomness or unpredictability. In a stochastic system, the outcome is not deterministic, meaning that even if you start from the same conditions, the result can vary each time. This is because there are elements of chance or randomness influencing the system.

Examples to Explain Stochastic:

1. Rolling a Die:

• When you roll a die, you can’t predict with certainty which number will come up. The outcome is random, and each roll is independent of the previous ones. This randomness makes rolling a die a stochastic process.

1. Weather:

• Weather forecasts are an example of a stochastic system. Even with the best data and models, we can’t predict the weather with 100% certainty because there are many random factors at play, like small changes in wind or temperature that can have big effects.

1. Stock Market:

• The stock market is also stochastic. The price of stocks can fluctuate due to a variety of unpredictable factors, such as news events, investor sentiment, or changes in economic conditions.

Key Points:

• Randomness: Stochastic systems are characterized by randomness. The same action or input might lead to different results each time.
• Probability: Instead of certainty, outcomes in a stochastic system are described using probabilities. For example, you might know that there’s a 50% chance of rain tomorrow, but you can’t be sure if it will rain or not.
• Uncertainty: Stochastic processes involve uncertainty, making it impossible to predict exact outcomes, only likely ones.

In essence, “stochastic” is a term used to describe situations where outcomes are not fully predictable and where randomness plays a role.

When we say something is “stochastic,” it means that there is an element of randomness or unpredictability involved. In simpler terms, it means that the outcome is not entirely certain and can vary each time you try it, even if the situation appears to be the same.

Let’s break down the phrase:

“On stochastic tasks, actions must be repeated multiple times to obtain a reliable estimate of expected rewards.”

1. Stochastic Tasks: These are tasks where the outcome is not always the same, even if you do the same thing every time. For example, imagine flipping a coin. Even though the coin and the flipping action are the same each time, you might get heads sometimes and tails other times. This makes the task “stochastic” or random.
2. Actions and Repetition: Because the outcome can change each time, it’s not enough to try an action just once to see what happens. For example, if you’re trying to figure out the average number of times you’ll get heads when flipping a coin, you wouldn’t just flip the coin once; you’d flip it many times. This repetition helps you get a more accurate understanding of what to expect on average.
3. Reliable Estimate of Expected Rewards: Over many repetitions, you can start to see a pattern or an average outcome. For the coin example, after many flips, you might see that you get heads about 50% of the time. This average is your “expected reward” (in this case, how often you expect to get heads). By repeating the action multiple times, you get a more reliable (or trustworthy) estimate of what to expect.

In Summary:

When dealing with stochastic tasks (tasks with random outcomes), you need to perform actions multiple times to accurately understand what kind of results you can expect on average. This helps in making better decisions in the future, knowing that while individual outcomes can be random, there’s a reliable pattern when you look at the bigger picture.

Now Lets Understand Mathematically:-

From a mathematical point of view, “stochastic” refers to processes or systems that can be modeled using probability theory. In such processes, the outcome is not deterministic but instead described by a probability distribution, meaning that there are multiple possible outcomes, each with a certain likelihood.

Mathematical Explanation:

1. Stochastic Process:

• A stochastic process is a sequence of random variables indexed by time or space. Mathematically, a stochastic process is denoted as ({X_t}), where (t) is the index (often representing time), and (X_t) is the random variable at time (t).
• Example: A classic example of a stochastic process is a random walk. Suppose you start at position 0 on a number line. At each step, you either move one unit to the right (+1) or one unit to the left (-1) with equal probability. The position after (n) steps, denoted by (X_n), is a random variable whose value depends on the outcome of each step.

1. Probability Distributions:

• The behavior of stochastic processes is governed by probability distributions. For example, in a simple random walk, the probability of being at a certain position after a given number of steps can be described by a binomial distribution if the steps are independent and identically distributed.
• Example: If (X_n) represents the position after (n) steps in a random walk where each step is +1 or -1 with equal probability (0.5), the expected position (mean) after (n) steps is 0, but the variance grows with the number of steps, specifically as (n). The distribution of (X_n) after a large number of steps approximates a normal distribution due to the Central Limit Theorem.

1. Stochastic Differential Equations (SDEs):

• In more complex systems, stochastic processes are often modeled using stochastic differential equations, which are differential equations that include terms representing random noise.
• Example: The Black-Scholes equation used in financial mathematics to model the price of an option over time is a type of SDE. The equation has a deterministic part and a stochastic part (typically represented as (\sigma dW_t), where (W_t) is a Wiener process or Brownian motion).

Example in Detail:

Consider the random walk example mentioned above:

• Random Walk: Let (X_0 = 0). At each step (n), the position (X_n) is updated based on:
  [
  X_{n+1} = X_n + \epsilon_n
  ]
  where (\epsilon_n) is a random variable that takes the value +1 or -1 with equal probability (0.5 each).
• Expected Value: The expected value (E[X_n]) after (n) steps is 0, because each step has an equal chance of moving left or right.
• Variance: The variance (Var(X_n)) increases with the number of steps and is given by:
  
• [X_{n+1} = X_n + \epsilon_n]
  Since ( \epsilon_n ) has a variance of 1, the variance of (X_n) after (n) steps is (n).
• Probability Distribution: The distribution of (X_n) after a large number of steps approaches a normal distribution due to the Central Limit Theorem, with a mean of 0 and variance (n).

This random walk is a simple example of a stochastic process where outcomes evolve over time according to probabilistic rules, leading to a spread of possible outcomes that can be analyzed using statistical methods.

Stochastic vs. Deterministic:

In contrast, a deterministic process has no randomness involved. If you know the initial conditions and the rules governing the process, you can predict the outcome exactly. For example, in a deterministic process, if you move 1 unit to the right at every step starting from 0, you will always be at position (n) after (n) steps. No probability is involved.

In summary, mathematically, “stochastic” refers to processes where outcomes are governed by probability distributions, often requiring multiple realizations or trials to understand the expected behavior of the system over time.