7  Probability Distributions

Probability not only helps us understand how likely an event is to occur, but also forms the foundation of many statistical methods used for decision-making. When a process or experiment produces varying outcomes, we use a random variable to represent those outcomes and a probability distribution to describe how the probabilities are assigned to each possible value. Understanding the shape and properties of a distribution is essential because it determines how data behave, how we calculate probabilities, and how we make predictions. From distributions for continuous variables to the behavior of statistics such as sample means, probability distributions serve as the core of inferential statistics.

This material will guide you through several key concepts:

Each section is supported with video explanations to enhance conceptual understanding. By mastering these topics, you will be better equipped to analyze data, build statistical models, and draw conclusions based on solid probabilistic principles.

7.1 Continuous Random

Understanding these basics will provide a strong foundation as we transition into the main topic of this video: Continuous Random Variables and Their Probability Distributions.

Probability Distribution of Continuous Variables

To understand continuous random variables, it is essential to know how probability is represented using a Probability Density Function (PDF).
Unlike discrete random variables, a continuous random variable does not assign probability to individual points. Instead, probability is obtained from the area under the PDF curve.

7.1.1 Random Variable

A random variable is continuous if it can take any value within an interval on the real number line.
Examples include: height, time, temperature, age, pressure, and velocity.

Key characteristics:

  • The variable takes values in an interval such as \((a, b)\) or even \((-\infty, +\infty)\).
  • The probability of any single point is always zero:

\[ P(X = x) = 0 \]

  • Probabilities are meaningful only over intervals:

\[ P(a \le X \le b) = \int_a^b f(x)\,dx \]

7.1.2 Probability Density Funct.

A function \(f(x)\) is a valid Probability Density Function (PDF) if it satisfies:

1. Non-negativity

\[ f(x) \ge 0 \quad \forall x \]

2. Total Area Equals 1

\[ \int_{-\infty}^{\infty} f(x)\,dx = 1 \]

Interpretation:

  • Larger values of \(f(x)\) indicate higher probability density around that value.
  • However, \(f(x)\) is not a probability; probabilities come from the area under the curve.

Example PDF: \(f(x) = 3x^2\) on \([0,1]\)

Consider the probability density function:

\[ f(x) = 3x^2,\quad 0 \le x \le 1 \]

Validation:

\[ \int_0^1 3x^2\,dx = 1 \]

7.1.3 Probability on an Interval

To compute probability within an interval:

\[ P(a \le X \le b) = \int_a^b 3x^2\, dx \]

Example:

\[ P(0.5 \le X \le 1) \]

7.1.4 Cumulative Distribution Funct.

The Cumulative Distribution Function (CDF) is defined as:

\[ F(x) = P(X \le x) = \int_0^x 3t^2\, dt = x^3 \]

Relationship between PDF and CDF:

\[ f(x) = F'(x) \]

7.2 Sampling Distributions

Before exploring the concept of sampling distributions in detail, this video provides a clear visual explanation of how statistics such as sample means behave when repeatedly drawn from the same population. It offers an intuitive foundation for understanding variability, uncertainty, and why sampling distributions are essential in statistical inference. Please watch the video below before continuing with the material.

Sampling Distributions

7.3 Central Limit Theorem

Central Limit Theorem

7.4 Sample Proportion

Sampling Distribution of the Sample Proportion

7.5 Review Sampling Distribution

Review Sampling Distribution

References