Understanding Joint Distribution of Random Variables in Excel

A joint distribution of random variables is a way of describing how the probabilities of two or more variables are related. For example, if we have two random variables X and Y that represent the outcomes of rolling two dice, we can use a joint distribution to show the probability of getting any combination of values for X and Y.

One way to represent a joint distribution is with a table that lists all the possible values of X and Y, and the corresponding probabilities. For example, the table below shows the joint distribution of X and Y for two fair dice:

The table shows that the probability of getting any pair of values for X and Y is equal to 1/36, since there are 36 possible outcomes and each one is equally likely. We can write this as P(X = x, Y = y) = 1/36 for any x and y between 1 and 6.

A joint distribution can also be used to study the relationship between two random variables, such as whether they are independent or dependent. Two random variables are independent if the probability of one does not depend on the value of the other. For example, if we roll two fair dice, the value of X does not affect the value of Y, and vice versa. We can check this by looking at the joint distribution table or formula, and seeing if the probability of X and Y is equal to the product of the probabilities of X and Y separately. For example, P(X = 1, Y = 2) = 1/36 = P(X = 1) * P(Y = 2) for two fair dice, so X and Y are independent.

Two random variables are dependent if the probability of one does depend on the value of the other. For example, if we draw two cards from a deck without replacement, the value of X (the first card) affects the value of Y (the second card), and vice versa. We can check this by looking at the joint distribution table or formula, and seeing if the probability of X and Y is not equal to the product of the probabilities of X and Y separately. For example, P(X = Ace, Y = Ace) = 1/221 ≠ P(X = Ace) * P(Y = Ace) for two cards without replacement, so X and Y are dependent.

Basic Theory:

A joint distribution involves the simultaneous occurrence of two or more random variables. It describes the probabilities associated with different combinations of values for these variables. For two random variables X and Y, the joint distribution function is denoted as P(X = x, Y = y). The marginal distributions of X and Y can be obtained by summing or integrating over all possible values of the other variable.

Procedures in Excel:

1. Data Setup:
   • Organize your data into a table with columns for each random variable.
   • Each row represents a different observation or scenario.
2. Calculate Joint Probabilities:
   • Use Excel formulas to calculate the joint probabilities for each combination of values.
   • If your data is discrete, use COUNTIFS or SUMIFS functions.
   • If continuous, consider using Excel’s integration functions or construct a frequency distribution.
3. Marginal Distributions:
   • Sum or integrate the joint probabilities to obtain the marginal probabilities for each variable.
   • Ensure that the sum of probabilities equals 1 for both variables.
4. Create a Joint Probability Table:
   • Build a table to display the joint probabilities and marginal probabilities.
5. Visualization (Optional):
   • Use Excel charts (scatter plots, bar charts) to visualize the joint distribution.

Comprehensive Example Scenario:

Consider a scenario where X represents the number of products sold in a day and Y represents the profit made. The joint distribution table might look like this:

Calculations:

1. Marginal Distributions:
   • Sum the probabilities for each value of X and Y.
2. Expected Values:
   • Calculate the expected values of X and Y.
3. Covariance and Correlation:
   • Use Excel functions like COVARIANCE.P and CORREL to quantify the relationship between X and Y.

Result of the Scenario:

After performing the calculations, we find that the expected number of products sold (E[X]) is 2, the expected profit (E[Y]) is $70, the covariance (cov(X,Y)) is 5, and the correlation coefficient (ρ(X,Y)) is 0.5.

Other Approaches:

1. Simulation in Excel:
   • Utilize Excel’s RAND function to simulate random variables based on their probabilities.
2. Advanced Statistical Functions:
   • Explore Excel’s statistical functions like LINEST for linear regression on joint distributions.