Problem 7 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Consider the joint distribution of Quick exercise \(9.2\) and take \(\varepsilon\) fixed between \(-1 / 4\) and \(1 / 4\) : $$ \begin{array}{cccc} & {b} & \\ a & 0 & 1 & p_{X}(a) \\ \hline 0 & 1 / 4-\varepsilon & 1 / 4+\varepsilon & 1 / 2 \\ 1 & 1 / 4+\varepsilon & 1 / 4-\varepsilon & 1 / 2 \\ \hline D_{Y}(b) & 1 / 2 & 1 / 2 & 1 \end{array} $$ a. Take \(\varepsilon=1 / 8\) and compute \(\operatorname{Cov}(X, Y)\). b. Take \(\varepsilon=1 / 8\) and compute \(\rho(X, Y)\). c. For which values of \(\varepsilon\) is \(\rho(X, Y)\) equal to \(-1,0\), or 1 ?

Step-by-Step Solution

Answer

a. Cov(X, Y) = 0 b. ρ(X, Y) = 0 c. ρ(X, Y) = 0 for any ε within the range.

Step 1: Compute Joint Probabilities

With \( \varepsilon = \frac{1}{8} \), calculate the joint probabilities from the table:- \( P(X=0, Y=0) = \frac{1}{4} - \varepsilon = \frac{1}{8} \)- \( P(X=0, Y=1) = \frac{1}{4} + \varepsilon = \frac{3}{8} \)- \( P(X=1, Y=0) = \frac{1}{4} + \varepsilon = \frac{3}{8} \)- \( P(X=1, Y=1) = \frac{1}{4} - \varepsilon = \frac{1}{8} \)

Step 2: Compute Marginal Probabilities

Calculate the marginal probabilities of \( X \) and \( Y \):- \( P_X(0) = \frac{1}{2} \), \( P_X(1) = \frac{1}{2} \)- \( P_Y(0) = \frac{1}{2} \), \( P_Y(1) = \frac{1}{2} \)

Step 3: Calculate Expected Values

Compute the expected values:- \( E[X] = 0 \times \frac{1}{2} + 1 \times \frac{1}{2} = \frac{1}{2} \)- \( E[Y] = 0 \times \frac{1}{2} + 1 \times \frac{1}{2} = \frac{1}{2} \)

Step 4: Calculate Covariance

Calculate the covariance:\[\operatorname{Cov}(X, Y) = E[XY] - E[X]E[Y]\]Where:\[E[XY] = 0 \times 0 \times \frac{1}{8} + 0 \times 1 \times \frac{3}{8} + 1 \times 0 \times \frac{3}{8} + 1 \times 1 \times \frac{1}{8} = \frac{1}{8}\]Thus, \( \operatorname{Cov}(X, Y) = \frac{1}{8} - \frac{1}{2}\times\frac{1}{2} = 0 \).

Step 5: Calculate Correlation (ρ)

With \( \operatorname{Cov}(X, Y) = 0 \), the correlation \( \rho(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y} = 0 \) since the standard deviations are both \( \frac{1}{2} \), so \( \rho(X, Y) = \frac{0}{\frac{1}{2} \times \frac{1}{2}} = 0 \).

Step 6: Determine ε Values for Specific Correlations

The correlation \( \rho(X, Y) \) equals \(-1, 0,\) or \(1\) when:- \( \rho = 0 \) for any \( \varepsilon \) since the covariance is zero indicating independence.- \( \rho = \pm 1 \) would require that the variance be entirely explained by the covariance, which does not apply for the given table values for any \( \varepsilon \) in \(-\frac{1}{4}, \frac{1}{4}\). Therefore, there are no values of \( \varepsilon \) that make \( \rho(X, Y) \) equal to \(-1\) or \(1\) within the specified range.

Key Concepts

Joint Probability Distribution

A joint probability distribution is a mathematical concept that helps us understand the likelihood of different outcomes happening at the same time. Imagine you have two variables, say X and Y, which can each take on certain values. The joint probability distribution provides the probability that X and Y both happen to take on specific values simultaneously.

This distribution is typically presented in a table or a matrix form. In our original problem, it gives the probabilities like \(P(X=0, Y=0)\) or \(P(X=1, Y=1)\). Here, you see how likely each combo of X and Y is to occur.

• Helps in finding correlations by giving a full picture of the relation between two variables.
• Essential for calculating other statistical measures like covariance.

By analyzing this table, we gain insights into whether there's any relationship between X and Y and how strong that relationship might be.

Marginal Probability

Marginal probability refers to the probability of a single event occurring without considering the other event. Simple, right? It basically just means ignoring the joint part and focusing on one variable. This is useful for general insights about each variable.

To get these probabilities, sum up the probabilities across rows or columns in the joint distribution table. In our problem, for instance, the marginal probability \(P_X(0)\) refers to the total probability that X equals 0, regardless of Y, which is calculated by adding \(P(X=0, Y=0)\) and \(P(X=0, Y=1)\).

• Vital for understanding the overall likelihood of a single outcome occurring.
• Helps in breaking down complex joint probabilities to simpler forms.

It filters out specific events, making calculations and understanding simpler to remember and utilize.

Expected Value

The expected value is a concept that sounds fancier than it is. It’s basically the average outcome you would "expect" from repeating an experiment or a process many, many times. For any random variable, it's calculated as the sum of all possible values, each multiplied by their probability of occurring.

In our case, this involves calculating separate expected values for X and Y. You multiply each value by its marginal probability and then add up all those products. It gives us a kind of "center of gravity" of the variable's probability distribution.

• Acts like a weighted average, giving more weight to more likely outcomes.
• Helps in predicting future results and making informed decisions.

By knowing the expected values, you get a glimpse of the general trend or direction of your data.

Correlation Coefficient

The correlation coefficient, often denoted as \( \rho \), measures the strength and direction of a linear relationship between two random variables. It's intriguing because it quantifies how much two variables move together, from \(-1\) to \(1\).

In layman terms, a correlation of ±1 indicates perfect linear relationship (either positive or negative). Meanwhile, a zero correlation suggests no linear relationship at all. To calculate it, you need the covariance and the standard deviations of the variables. If the covariance is zero, as in our example, the correlation is zero, highlighting independence of the variables.

• Extremely practical for identifying relationships between datasets.
• A pivotal statistic in regression analysis and hypothesis testing.

Remember, correlation doesn’t imply causation. But, it’s a fabulous tool for spotting patterns and relationships within your data.