Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Consider the joint probability distribution of the discrete random variables \(X\) and \(Y\) from the Melencolia Exercise 9.1. Compute \(\operatorname{Cov}(X, Y)\). $$ \begin{array}{ccccc} \hline \hline & {a} \\ b & 1 & 2 & 3 & 4 \\ { 2 - 5 } & 16 / 136 & 3 / 136 & 2 / 136 & 13 / 136 \\ 2 & 5 / 136 & 10 / 136 & 11 / 136 & 8 / 136 \\ 3 & 9 / 136 & 6 / 136 & 7 / 136 & 12 / 136 \\ 4 & 4 / 136 & 15 / 136 & 14 / 136 & 1 / 136 \\ \hline \hline \end{array} $$

Step-by-Step Solution

Answer

\(\operatorname{Cov}(X, Y) = -0.2206\)

Step 1: Understand the Definition of Covariance

The covariance, \( \operatorname{Cov}(X, Y) \), of two random variables \(X\) and \(Y\) is given by \(\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]\). This can also be calculated using \(\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\).

Step 2: Find Marginal Probabilities

Sum across the rows to get \(P(X=x)\) and sum down the columns to get \(P(Y=y)\). These are required to determine the expectations \(E[X]\) and \(E[Y]\).

Step 3: Calculate \(E[X]\) and \(E[Y]\)

Compute the expected value of \(X\), \(E[X] = \sum_x x \cdot P(X=x)\) and for \(Y\), \(E[Y] = \sum_y y \cdot P(Y=y)\) using the marginal probabilities found in Step 2.

Step 4: Calculate \(E[XY]\)

Calculate the expectation \(E[XY] = \sum_x \sum_y x \cdot y \cdot P(X=x, Y=y)\). Use the joint probabilities given in the table.

Step 5: Compute \(\operatorname{Cov}(X, Y)\)

Substitute \(E[XY]\), \(E[X]\), and \(E[Y]\) into the formula \(\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\) to find the covariance.

Key Concepts

Discrete Random Variables

Discrete random variables are a fundamental concept in probability and statistics. A discrete random variable is one that can take on a finite or countably infinite set of distinct values. Unlike continuous random variables, which can take on any value within an interval, discrete random variables deal with specific, distinct outcomes.
For example, the result of rolling a die is a discrete random variable because it can only be one of the six possible outcomes (1 through 6). Moreover, in a probabilistic scenario, each outcome is associated with a specific probability.
When analyzing problems involving discrete random variables, we sum the probabilities of each possible outcome to ensure they total to 1. This property guarantees that the random variable covers all possible events.
Discrete random variables are often paired with probability distributions, such as the joint probability distribution, which describes how two or more random variables relate to each other. Understanding discrete random variables lays the groundwork for evaluating more complex statistical concepts like covariance.

Marginal Probabilities

Marginal probabilities play a crucial role in understanding the behavior of individual random variables within a joint probability distribution. The marginal probability of a discrete random variable is found by summing the joint probabilities across rows or down columns of the probability distribution table.
For instance, to find the marginal probability of random variable \(X\), we sum over all values of \(Y\) in the joint distribution. Conversely, when finding the marginal probability of \(Y\), sum over all values of \(X\).
These probabilities help compute expected values, a key step in calculating covariance.
In the exercise, marginal probabilities allow us to assess the likelihood of a single variable occurring without considering the influence of other variables. By focusing on one variable, we can simplify the analysis to examine its contribution to the overall probability model.

Joint Probability Distribution

A joint probability distribution is a statistical model that describes the probability of two or more discrete random variables occurring simultaneously. It is presented in the form of a table where the intersection of rows and columns represents the joint probabilities of the respective random variable pairs.
Joint distributions offer a comprehensive understanding of how random variables interact, showing probabilities for every possible combination of their values. This is essential when evaluating relationships such as covariance.
For example, in the context of the exercise, the joint probability table shows the likelihood of specific \(X, Y\) outcomes. Each cell of the table holds the probability of a paired outcome occurring.
By examining the joint distribution, we gain insights into potential dependencies or correlations between variables, which are pivotal in calculating statistics such as the expected value of their product, \(E[XY]\), a necessary component for computing covariance.

Expected Value

The expected value is a critical concept in probability, offering a measure of the "center" or "average" of a random variable's possible outcomes, weighted by their probabilities. For discrete random variables, it is calculated as a sum of each outcome multiplied by its probability.
In simpler terms, the expected value gives us an idea of what to expect in the long run (or over many trials) from a random variable. For example, if you were to roll a fair die over many rolls, the expected value would be 3.5, even though you can never roll a 3.5 on a six-sided die.
For a pair of random variables, not only can we compute their individual expected values, \(E[X]\) and \(E[Y]\), but also their combined expectation, \(E[XY]\), using marginal and joint probabilities.
This calculation becomes especially important in determining covariance, where it contributes to finding out if and how two variables may move together, offering insight into their relationship.