# Random Variable

A random variable, often denoted as X, is a variable that represents the potential numerical outcomes of a random event or process. Random variables come in two main types: discrete and continuous.

### Discrete Random Variable:

A discrete random variable is one that can only assume a countable number of distinct values. These values are typically whole numbers and can be listed, such as 0, 1, 2, 3, 4, and so on. Discrete random variables are commonly associated with situations involving counts or whole units. If a random variable can take only a finite number of distinct values, it is categorized as discrete. Examples of discrete random variables include:

• The number of children in a family.
• The attendance on a Friday night at a cinema.
• The count of patients in a doctor’s surgery.
• The number of defective light bulbs in a box of ten.

### Continuous Random Variable:

A continuous random variable is one that can take on an infinite number of possible values. These values form a continuous range, often involving measurements with decimal points. Continuous random variables are frequently used to represent quantities that can vary across a broad spectrum. Examples of continuous random variables include:

• Height of individuals.
• Weight of objects.
• The amount of sugar in an orange, which can take any value within a range.
• The time required to run a mile, which can be measured with high precision.

NSE price changes in 0.05

We are referring to a specific example of a continuous random variable, where the random variable represents the changes in the National Stock Exchange (NSE) prices, and the typical change is 0.05 units. This represents a continuous random variable as it can take on an infinite number of possible price change values.

Stock prices or values are discrete and Stock market returns are continuous variables.*

### Stock Prices: A Discrete World

The heart of any stock market is, undeniably, stock prices. These prices represent the current value of a company’s shares at any given moment. In the context of INR, it’s important to understand that stock prices are inherently discrete.

They move in predefined tick levels, each step representing a specific INR denomination. For instance, you might observe a stock trading at ₹100, ₹100.05, ₹100.10, and so forth. This discrete movement is in line with the nature of financial markets, where stocks are bought and sold in whole units.

### Stock Market Returns: A Continuous Perspective

Stock market returns are often calculated using formulas like `(Current Price - Previous Price) / Previous Price`. The outcome of this calculation is a continuous variable that reflects the rate of change, expressed as a percentage. While the underlying stock prices may be discrete, the returns they yield, especially when viewed as percentages, span a continuous spectrum of values.

Furthermore, when financial professionals discuss returns in a more holistic, long-term perspective, they often employ continuously compounded returns. These returns are articulated as logarithmic returns and are symbolized as `ln(T/P)`, where ‘T’ signifies the final price, and ‘P’ represents the initial price.

As Log(T/P) is a continuous variable; stock market returns are continuous. It factors in the compounding effect achieved through reinvesting gains over time.

In practical terms, continuously compounded returns offer a more precise depiction of how investments grow over extended periods, as they account for the continuous, compounding nature of returns.

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