Understanding Key Types of Probability Distributions in Statistics

Core Concepts in Probability Distributions

Random Variables Defined

Random variables are those variables that assume different values as a result of a random experiment. They are classified into two main types:

• Discrete Random Variables: These support only integer values (countable outcomes).
• Continuous Random Variables: These support all types of values, including integers and fractions (measurable outcomes).

What is a Discrete Probability Distribution?

It is a theoretical frequency distribution, similar to relative frequencies. It maps the values of variables to the probability of occurrence of that value, representing what would be expected if the experiment were carried out repeatedly.

Key Discrete Probability Distributions

Binomial Distribution Characteristics

The Binomial Distribution is a discrete probability distribution based on two mutually exclusive outcomes: success ($P$) and failure ($Q$). It consists of two components: a combinatorial component and a probabilistic component.

Key Properties

• Based on two outcomes: success ($P$) and failure ($Q$).
• Used for infinite populations or sampling with replacement.
• Each trial result is statistically independent.
• The probability of success ($P$) and failure ($Q$) remains constant from trial to trial.

Hypergeometric Distribution Characteristics

The Hypergeometric Distribution is a discrete probability distribution where the random variable ($x$) accepts only integer values.

Key Properties (Contrast with Binomial)

• Used for finite populations or sampling without replacement.
• The probability of success is not constant; it is dependent on previous trials.
• It is applicable when two or more results can occur within the finite sample.

Poisson Distribution and Applications

The Poisson Distribution is a discrete probability distribution often used when the number of trials ($n$) is large. It describes the number of times an event is expected to occur within a specified interval (such as time, distance, area, or volume).

The Normal (Continuous) Distribution

Key Properties of the Normal Distribution

The Normal Distribution is a fundamental continuous probability distribution.

• It is perfectly symmetric.
• It is asymptotic (the tails approach the horizontal axis but never touch).
• The total area under the curve equals 1 (or 100%).
• The mean, median ($M$), and mode are all located at the center of the distribution.