Homework 4

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Statistical Independence

Statistical independence is a fundamental concept in probability theory and statistics, describing a relationship between two events or random variables where the occurrence of one does not affect the occurrence of the other. This concept is crucial for building probabilistic models and analyzing data across various fields.

Defining Statistical Independence

Two events, say $A$ and $B$, are considered statistically independent if the occurrence of one does not influence the likelihood of the other happening. This independence can be formally defined using probabilities in several equivalent ways.

Definition

Events $A$ and $B$ are independent if and only if:

\[P(A \cap B) = P(A) \cdot P(B)\]

where:

$P(A \cap B)$ is the probability that both events $A$ and $B$ occur together (joint probability)
$P(A)$ and $P(B)$ are the individual probabilities of $A$ and $B$ occurring (marginal probabilities)

Understanding the Formula

The formula $P(A \cap B) = P(A) \cdot P(B)$ implies that if $A$ and $B$ are independent, the likelihood of both events occurring together is simply the product of their individual probabilities. This means that knowing one event has happened provides no information about the probability of the other event happening.

Independence Through Conditional Probability

Another equivalent way to define independence uses conditional probability. The conditional probability of $A$ given $B$, written $P(A|B)$, is the probability of $A$ occurring given that $B$ has already occurred.

For $A$ and $B$ to be independent, the occurrence of $B$ should not change the probability of $A$. Therefore, if $A$ and $B$ are independent:

\[P(A|B) = P(A)\]

and equivalently:

\[P(B|A) = P(B)\]

This means that knowing $B$ has occurred does not affect the likelihood of $A$, and vice versa, reinforcing that $A$ and $B$ are independent.

Independence for Random Variables

The concept extends naturally to random variables. For discrete random variables $X$ and $Y$, independence means that the joint probability distribution is the product of their marginal distributions:

\[P(X = x \text{ and } Y = y) = P(X = x) \cdot P(Y = y)\]

for all possible values $x$ and $y$.

For continuous random variables, independence requires that the joint probability density function $f_{X,Y}(x,y)$ equals the product of the marginal density functions:

\[f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)\]

for all values of $x$ and $y$.

Comparing to Dependent Events

In contrast, if two events are dependent, the occurrence of one event affects the probability of the other. For dependent events $A$ and $B$, we would observe that:

$P(A \cap B) \neq P(A) \cdot P(B)$
$P(A|B) \neq P(A)$
$P(B|A) \neq P(B)$

This deviation from the independence condition indicates that information about one event provides insight into the likelihood of the other.

Examples and Applications

Example 1: Rolling Two Dice

Consider rolling two six-sided dice, one green and one red. Let:

Event $A$ be “the green die lands on 4”
Event $B$ be “the red die lands on 6”

The probability of $A$ is $P(A) = \frac{1}{6}$, and the probability of $B$ is $P(B) = \frac{1}{6}$. Because the outcome of one die does not influence the outcome of the other, the probability of both events happening together is:

\[P(A \cap B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\]

This confirms that the events are independent: knowing that one die shows a particular number doesn’t tell us anything about the other.

Example 2: Coin Flips and Die Rolls

Consider flipping a fair coin and rolling a fair die simultaneously. Let:

Event $A$: The coin shows heads
Event $B$: The die shows a 3

Since $P(A) = \frac{1}{2}$ and $P(B) = \frac{1}{6}$, and these events are independent:

\[P(A \cap B) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}\]

Summary

Statistical independence is a fundamental concept that describes situations where events or variables do not influence each other. The formal mathematical definitions through joint probabilities and conditional probabilities provide precise tools for identifying and working with independent events. Understanding this concept is essential for proper statistical analysis and probabilistic reasoning, as it allows for simplified calculations and forms the foundation for many statistical methods and models.

Raffaele Ruggeri