When dealing with random variables in probability and statistics, understanding how their properties combine is essential. The question asks about the distribution of the sum of two normally distributed random variables, X and Y.

Normal Distribution Fundamentals

A normal distribution, often called a Gaussian distribution, is a continuous probability distribution that is symmetric around its mean, creating a distinctive bell-shaped curve. It is widely used in statistics because it naturally models many phenomena. A normal distribution is completely defined by its mean (\(\mu\)) and its standard deviation (\(\sigma\)).

Sum of Normal Variables Rule

One of the key properties concerning normal distribution is how sums of such variables behave. Here's the rule:

• If X and Y are two normally distributed random variables, their sum, \(S = X + Y\), is also normally distributed if and only if X and Y are independent.
• When they are independent, the mean of the sum is the sum of their individual means: \(\mu_S = \mu_X + \mu_Y\).
• The variance of the sum is the sum of their individual variances: \(\sigma_S^2 = \sigma_X^2 + \sigma_Y^2\). This property for variance holds specifically under the condition of independence.

This means that if X and Y are independent, their sum will always have a normal distribution, regardless of their individual means or standard deviations.

Independence in Statistics

Independence between two random variables means that the outcome or value of one variable does not provide any information about the outcome or value of the other variable. In the context of normal distribution, if X and Y are dependent, their sum might not necessarily be normally distributed. For example, if \(Y = -X\), and X is normally distributed, then \(X+Y = X-X = 0\), which is a constant and not a normal distribution.

Analyzing the Options for Sum of Normal Variables

Let's examine why the other options provided are not universally correct:

• Option 1: always normally distributed
  This statement is too strong and incorrect. As discussed, if X and Y are dependent, their sum is not guaranteed to be normally distributed. The critical condition of independence is missing here.
• Option 3: normally distributed, only if X and Y have the same standard deviation
  This is incorrect. The normality of the sum of two independent normal random variables does not depend on them having identical standard deviations. They can have different standard deviations, and their sum will still be normal as long as they are independent.
• Option 4: normally distributed, only if X and Y have the same mean
  This is also incorrect. Similar to the standard deviation, the means of X and Y do not need to be the same for their sum to be normally distributed. The sum of two independent normal variables remains normal regardless of whether their means are equal or different.

Therefore, the defining condition for the sum of two normally distributed random variables to itself be normally distributed is that the variables X and Y must be independent.