What distribution is produced when subtracting one IID normally distributed variable from another, both having a mean of 1 and a variance of 3?

This inquiry pertains to the realm of statistics, particularly focusing on probability distributions. Key concepts to grasp in this question include Normal Distribution, Independent Identically Distributed (IID) variables, and the outcome when one variable is subtracted from another. Normally Distributed Variables: A variable is considered to be normally distributed if its distribution aligns with a specific bell-shaped curve (Gaussian distribution). Each normally distributed variable is characterized by two parameters: the mean and the variance. In this scenario, both variables have a mean of 1 and a variance of 3. Independent Identically Distributed (IID): This indicates that all variables are the same (they adhere to the same probability distribution and possess identical parameters) and independent (the result of one variable does not affect the result of another variable). You are required to determine the resulting distribution when subtracting two of these IID normally-distributed variables. In other words, what will be the mean and variance of the new variable/distribution that arises from the subtraction?

When two independent random variables that are normally distributed are subtracted, the outcome is also a normally distributed random variable. The mean (μ) of this new distribution is the difference between the means of the original variables, while the variance (σ^2) is the sum of the variances of the original variables. In this scenario, both original variables have a mean (μ) of 1 and a variance (σ^2) of 3. Consequently, the resulting distribution after the subtraction will have a mean of 1 - 1 = 0 and a variance of 3 + 3 = 6. This results in a Normal distribution with a mean of 0 and a variance of 6.

Independent and identically distributed (IID) normal variables remain normal when one is subtracted from the other. The principle states that the difference between two independent normal random variables is also a normal random variable. The mean of the resultant distribution is calculated as the algebraic difference of the individual means. Given that both variables have a mean of 1, the resulting mean would be 1 - 1 = 0. The variance of the resulting distribution is determined by summing the variances of the two variables. Since each variable has a variance of 3, the total variance when subtracting them will be 3 + 3 = 6. Therefore, the distribution resulting from the subtraction of these two IID normally distributed variables will be a normal distribution with a mean of 0 and a variance of 6.