Problem 59 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(X\) and \(Y\) are discrete random variables with the joint pdf $$ \begin{array}{cc} \hline(x, y) & f x, Y(x, y) \\ \hline(1,2) & \frac{1}{2} \\ (1,3) & \frac{1}{4} \\ (2,1) & \frac{1}{8} \\ (2,4) & \frac{1}{8} \\ \hline \end{array} $$ Find the correlation coefficient between \(X\) and \(Y\).

Step-by-Step Solution

Answer

The correlation coefficient will be calculated using the four steps outlined above. The answer will be a value between -1 and +1, where the sign expresses the direction of the relation between \(X\) and \(Y\), and the absolute value expresses the strength of that relationship.

Step 1: Calculate the Mean

First, calculate the means for both \(X\) and \(Y\). The mean of a discrete random variable is computed as \(\mu = \Sigma {xf(x)}\). Use the provided data pairs and their probabilities to calculate the mean for both \(X\) and \(Y\).

Step 2: Calculate the Variance

Next, calculate the variances for \(X\) and \(Y\). Variance represents the dispersion of the random variables around their mean and is calculated as \(\sigma^2 = \Sigma [x^2f(x)] - \mu^2\). Again, use the pairs and their probabilities to find individual variances of \(X\) and \(Y\).

Step 3: Calculate the Covariance

Then, compute Cov(X,Y), the covariance between X and Y. Covariance measures how much \(X\) and \(Y\) vary together. It's calculated as \(COV(X, Y) = E[XY] - E[X]E[Y]\). Use the information given to calculate this value.

Step 4: Compute the Correlation Coefficient

Lastly, calculate the correlation coefficient, denoted by \(\rho\). The Correlation Coefficient is computed as \(\rho = \frac{COV(X,Y)}{\sigma_X \sigma_Y}\), where \(\sigma_X\) and \(\sigma_Y\) are the standard deviations of \(X\) and \(Y\) respectively. This is calculated by taking the square root of variance.

Key Concepts

Joint Probability Distribution Function

Understanding the joint probability distribution function (pdf) is essential in statistics, especially when dealing with two or more discrete random variables. Joint pdf is a function that provides the probability that each of the random variables falls within a particular range or specific values. In other words, it tells us how likely it is for a combination of random variables to occur together.

For discrete random variables like in our exercise, the joint pdf is represented as a table or a formula that assigns a probability to each pair of outcomes for the two variables, \(X, Y\). If you have the joint pdf, you can find out the probability of any combination of \(X\) and \(Y\) occurring by looking at the value assigned to that pair in the function.

Mean of Discrete Random Variables

The mean, or expected value, of a discrete random variable is a measure of its central tendency — essentially, it tells you where most of the values are expected to fall on average. To calculate the mean of a discrete random variable, you multiply each possible value the variable can take by its probability and then sum all these products together.

In the given exercise, this would be done by using the formula \(\mu = \Sigma {x\cdot f(x)}\), where \(x\) is a value that the random variable can assume, and \(f(x)\) is the probability of \(x\) occurring. The outcome yields the average or mean value for the variable.

Variance and Standard Deviation

Variance and standard deviation are two foundational concepts in statistics that measure the spread of a set of data points. Variance, denoted by \(\sigma^2\), is the average of the squared differences from the mean, providing a numerical value that describes how much the values of a random variable differ from the mean. The formula for variance is \(\sigma^2 = \Sigma [(x - \mu)^2 \cdot f(x)]\), where \(\mu\) is the mean of the random variable.

The standard deviation is simply the square root of the variance and represents the average distance of each data point from the mean in original units, making it easier to interpret. Mathematically, standard deviation is expressed as \(\sigma = \sqrt{\sigma^2}\). Both variance and standard deviation give us insight into the variability of the random variable, with lower values indicating less variability.

Covariance

Covariance is a measure used to determine the relationship between two random variables, specifically how they change together. It indicates the direction of the linear relationship between variables. If the covariance is positive, it means that when one variable increases, the other is likely to increase as well. Conversely, a negative covariance indicates that when one variable increases, the other is likely to decrease.

To calculate covariance between two variables \(X\) and \(Y\), one would use the formula \(COV(X, Y) = E[XY] - E[X]E[Y]\), where \(E[XY]\) is the expected value of the product of \(X\) and \(Y\), while \(E[X]\) and \(E[Y]\) are the expected values or means of \(X\) and \(Y\) respectively. In the context of our exercise, calculating the covariance between \(X\) and \(Y\) involves using the provided joint probabilities to find \(E[XY]\) and then subtracting the product of their means.