Random Variables and Probability Distributions

Random Variables: Definition and Types (Discrete, Continuous)

Definition of a Random Variable

In the study of probability, we often deal with outcomes of experiments that are not inherently numerical (e.g., heads or tails when tossing a coin, types of defects in manufacturing, colours of cars). However, to perform mathematical and statistical analysis on these outcomes, it is convenient to associate them with numerical values.

A **Random Variable** is a concept that provides a way to assign a specific numerical value to each possible outcome of a random experiment. It is not a variable in the algebraic sense (where it represents an unknown value), but rather a variable whose value is determined by the outcome of a random process.

Formally, a random variable, usually denoted by a capital letter like $X$, $Y$, or $Z$, is a **function** that maps each outcome in the sample space ($S$) of a random experiment to a unique real number.

Where $S$ is the sample space of the random experiment, and $\mathbb{R}$ is the set of real numbers. For each outcome $s \in S$, the random variable $X$ assigns a specific numerical value $X(s)$. The set of all possible numerical values that the random variable can take is called its **range**.

Example: Experiment: Tossing two fair coins simultaneously.

The sample space is $S = \{HH, HT, TH, TT\}$.

Let the random variable $X$ be the "number of heads obtained" in this experiment.

• For the outcome $s = HH$, the number of heads is 2. So, $X(HH) = 2$.
• For the outcome $s = HT$, the number of heads is 1. So, $X(HT) = 1$.
• For the outcome $s = TH$, the number of heads is 1. So, $X(TH) = 1$.
• For the outcome $s = TT$, the number of heads is 0. So, $X(TT) = 0$.

The set of possible values (the range) that the random variable $X$ can take is $\{0, 1, 2\}$. The random variable transforms the non-numerical outcomes (HH, HT, etc.) into numerical outcomes (0, 1, 2).

Types of Random Variables

Random variables are categorized into two main types based on the set of values they can assume:

1. Discrete Random Variable:

   A discrete random variable is a random variable whose possible values are countable. This means the variable can take on either a finite number of distinct values or a countably infinite number of distinct values.

   • The values of a discrete random variable often arise from **counting** processes.
   • There are distinct gaps between the possible values that the variable can take.

   Examples:

   • The number of heads in 3 coin tosses (possible values: 0, 1, 2, 3 - finite number).
   • The number of defective items in a sample of 10 items from a production line (possible values: 0, 1, 2, ..., 10 - finite number).
   • The sum of the numbers when rolling two dice (possible values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 - finite number).
   • The number of phone calls received by a customer service center in an hour (possible values: 0, 1, 2, 3, ... - countably infinite number).

2. Continuous Random Variable:

   A continuous random variable is a random variable that can take on any value within a given interval or range on the real number line. These values are infinite and uncountable.

   • The values of a continuous random variable typically arise from **measuring** processes.
   • Between any two possible values, there are infinitely many other possible values.

   Examples:

   • The height of a person (e.g., any value between 150 cm and 200 cm).
   • The temperature of a liquid (e.g., any value between 0°C and 100°C).
   • The time taken to complete a race (e.g., any value in seconds greater than 0).
   • The weight of a package (e.g., any value in kilograms greater than 0).

The distinction between discrete and continuous random variables is fundamental because the methods used to describe their probability distributions and calculate probabilities are different.

Probability Distribution: Definition and Properties (for Discrete Random Variables)

Definition

The **probability distribution** of a **discrete random variable** $X$ is a complete description of the probabilities associated with each possible value that the random variable can take. It tells us how the total probability of 1 is distributed among the different possible values of $X$.

A probability distribution can be presented in various forms, but it essentially provides a list or function of all possible values of $X$ and their corresponding probabilities.

Let the possible values that a discrete random variable $X$ can take be $x_1, x_2, x_3, \dots$, where the sequence of values may be finite or countably infinite. The probability distribution assigns a probability to each of these values, denoted by $P(X=x_i)$, which is often represented by $p_i$.

Representations of a Discrete Probability Distribution

A probability distribution for a discrete random variable can be presented in the following ways:

1. Table Form:

This is the most common way to represent the probability distribution for a finite number of possible values. The table lists each possible value of the random variable $X$ and its corresponding probability:

Each $p_i$ must be calculated based on the underlying random experiment and the definition of the random variable.

2. Formula Form (Probability Mass Function - PMF):

For some discrete random variables, the probability $P(X=x)$ can be described by a mathematical formula, denoted as $p(x)$ or $f(x)$, which is called the **Probability Mass Function (PMF)**. The PMF gives the probability for each possible value $x$ of the random variable $X$.

3. Graphical Form:

A discrete probability distribution can be visualized using a bar chart or a line graph. The horizontal axis represents the possible values of the random variable ($x_i$), and the vertical axis represents the corresponding probabilities ($p_i$). The height of each bar (or the length of each line segment) indicates the probability of that specific value.

Properties of a Discrete Probability Distribution

For a table, formula (PMF), or graph to be considered a valid probability distribution for a discrete random variable $X$, it must satisfy the following two fundamental properties, derived from the axioms of probability:

1. Non-negativity:

   The probability of each possible value that the random variable can take must be greater than or equal to 0. Probabilities cannot be negative.

   $$P(X=x_i) = p_i \ge 0 \quad \text{for all possible values } x_i$$

   ... (2)

2. Summation to One:

   The sum of the probabilities of all possible values that the random variable can take must be exactly equal to 1. This reflects that the random variable must take on one of its possible values.

   $$\sum P(X=x_i) = \sum p_i = 1$$

   ... (3)

   The summation is over all possible values of $X$. For a countably infinite set of values, this is an infinite series that must converge to 1.

Any set of probabilities that satisfies these two properties constitutes a valid discrete probability distribution.

Example

Answer:

Probability Distribution (Applied Maths Definition)

From an applied mathematics or practical standpoint, a **probability distribution** is viewed as a **mathematical model** used to describe the behaviour of a random variable and quantify the likelihood of different possible outcomes for a random phenomenon observed in the real world. It goes beyond simply listing probabilities; it provides a structured way to represent and analyze uncertainty.

Key Aspects of Probability Distributions in Applied Contexts

• Modeling Randomness and Uncertainty:

  In applied fields, many quantities are inherently uncertain (e.g., the number of customers arriving per hour, the height of a randomly selected person, the outcome of a financial investment). A probability distribution provides a rigorous way to model this randomness by specifying the probabilities or probability densities associated with the various possible values the random variable can take. It replaces deterministic prediction with a probabilistic description.

• Foundation for Statistical Inference:

  Probability distributions are the bedrock of inferential statistics. We often assume that data collected from a sample comes from a population that follows a particular probability distribution (e.g., assuming heights are normally distributed). Based on this assumed distribution and the properties of the sample, we can make inferences and draw conclusions about the entire population.

• Decision Making Under Uncertainty:

  In areas like business, finance, engineering, and operations research, decisions must often be made despite uncertainty. Probability distributions are used to model uncertain inputs (like future demand, project completion time, machine failure rates, stock prices). Analyzing these distributions allows for risk assessment, calculation of expected costs or profits, and optimization of decisions based on probabilistic outcomes.

• Simulation and Modeling:

  Probability distributions are essential for simulating random processes on computers (e.g., Monte Carlo simulations). By drawing random numbers that follow a specific distribution, we can mimic real-world variability and study the long-term or probabilistic behaviour of complex systems.

• Summarizing and Characterizing Data:

  Beyond the full distribution, summary measures derived from it (like the expected value or mean, and the variance or standard deviation) are used to concisely describe the central tendency and spread of the random variable. Higher moments can describe shape characteristics like skewness and kurtosis.

• Fitting Theoretical Distributions:

  A key task in applied statistics is often to determine which standard theoretical probability distribution (e.g., Bernoulli, Binomial, Poisson, Uniform, Exponential, Normal) best fits an observed dataset. The choice depends on the nature of the variable (discrete or continuous) and the process believed to be generating the data. Once a suitable distribution is identified, its known properties can be used for further analysis and prediction.

In summary, from an applied perspective, probability distributions are not just abstract mathematical constructs but practical tools for modeling, analyzing, and making decisions in situations involving randomness and uncertainty, bridging the gap between theoretical probability and real-world data analysis.