9. Joint Probability: Concept, Formula, and Examples

This document explores the concept of joint probability, its relationship with conditional probability, and delves into probability density and distribution functions.

9.1 What is Joint Probability in Business Statistics?

Joint probability refers to the likelihood of two or more events occurring simultaneously. In business statistics, understanding joint probability is crucial for making informed decisions when multiple factors influence an outcome. It helps analyze scenarios where the occurrence of one event affects the probability of another.

Example: In a retail business, the joint probability of a customer purchasing a specific product and using a discount coupon.

9.2 Difference between Joint Probability and Conditional Probability

While related, joint probability and conditional probability are distinct:

• Joint Probability: The probability that both Event A and Event B occur. It is denoted as $P(A \cap B)$ or $P(A \text{ and } B)$.
• Conditional Probability: The probability that Event B occurs given that Event A has already occurred. It is denoted as $P(B|A)$.

The relationship between them is often expressed by the multiplication rule: $P(A \cap B) = P(A) * P(B|A)$ or $P(A \cap B) = P(B) * P(A|B)$

Key Difference: Joint probability deals with the co-occurrence of events, while conditional probability focuses on the probability of an event happening after another has already happened.

9.3 Probability Density Function: Meaning, Formula, and Graph

A Probability Density Function (PDF) is a function that describes the likelihood of a continuous random variable taking on a given value. For continuous variables, the probability of observing any single specific value is zero. Instead, the PDF assigns a probability density to each possible outcome, and the area under the curve between two points represents the probability that the variable falls within that range.

9.4 What is the Probability Density Function?

The PDF is a fundamental concept in the study of continuous probability distributions. It indicates where values are more likely to occur for a continuous random variable. The higher the value of the PDF at a particular point, the greater the likelihood of the random variable falling in a small interval around that point.

9.5 Probability Density Function Formula

For a continuous random variable $X$, its Probability Density Function, denoted as $f(x)$, must satisfy the following conditions:

1. Non-negativity: $f(x) \geq 0$ for all $x$.
2. Total Probability: The integral of the PDF over its entire domain must equal 1: $$ \int_{-\infty}^{\infty} f(x) dx = 1 $$

The probability that $X$ falls within a certain range $[a, b]$ is given by the integral of the PDF: $$ P(a \leq X \leq b) = \int_{a}^{b} f(x) dx $$

9.6 Properties of Probability Density Function

Key properties of PDFs include:

• Non-negativity: The function value is always greater than or equal to zero.
• Total Area is One: The area under the curve of the PDF across its entire range is exactly 1.
• Probability as Area: The probability of a random variable falling within a specific interval is represented by the area under the PDF curve within that interval.
• Probability of a Single Point is Zero: For a continuous random variable, $P(X=c) = 0$ for any specific value $c$.

9.7 Probability Distribution Function of Discrete Distribution

For a discrete random variable $X$, the probability distribution is described by its Probability Mass Function (PMF), often referred to as the Probability Distribution Function in the context of discrete distributions. The PMF assigns a probability to each distinct value that the random variable can take.

The PMF, denoted as $P(X=x)$, must satisfy:

1. Non-negativity: $P(X=x) \geq 0$ for all possible values of $x$.
2. Sum of Probabilities: The sum of probabilities for all possible values of $x$ must equal 1: $$ \sum_{x} P(X=x) = 1 $$

The probability of $X$ being less than or equal to a certain value $k$ is given by the Cumulative Distribution Function (CDF), denoted as $F(k)$: $$ F(k) = P(X \leq k) = \sum_{x \leq k} P(X=x) $$

Example (Discrete): Rolling a fair six-sided die. The PMF would be: $P(X=1) = 1/6$ $P(X=2) = 1/6$ ... $P(X=6) = 1/6$

9.8 Probability Distribution Function of Continuous Distribution

For a continuous random variable $X$, the probability distribution is described by its Probability Density Function (PDF), as discussed in sections 9.3-9.6. The term "Probability Distribution Function" for continuous variables almost always refers to the Cumulative Distribution Function (CDF).

The CDF, denoted as $F(x)$, gives the probability that the random variable $X$ will take on a value less than or equal to $x$.

$$ F(x) = P(X \leq x) $$

For a continuous random variable with PDF $f(t)$: $$ F(x) = \int_{-\infty}^{x} f(t) dt $$

Properties of CDF for Continuous Distributions:

• Range: $0 \leq F(x) \leq 1$ for all $x$.
• Monotonicity: $F(x)$ is non-decreasing.
• Limits:
  • $\lim_{x \to -\infty} F(x) = 0$
  • $\lim_{x \to \infty} F(x) = 1$
• Relationship with PDF: If $F(x)$ is differentiable, then $f(x) = \frac{d}{dx} F(x)$.

Example (Continuous): The height of adult males, often modeled by a normal distribution. The CDF would tell us the probability that a randomly selected adult male is shorter than a specific height.