Let X1 To X100 Be Iid N(μ, 1) Σ 12 Random Variables

1 Letx1x100be Iid Nμ1 Σ12 Random Variables It Is Not Necessary

1- Let X₁, ..., X₁₀₀ be independent and identically distributed (i.i.d.) random variables with a normal distribution, specifically N(μ₁, σ²₁). The question asks why it is not necessary to invoke the Central Limit Theorem (CLT) to conclude that their sum or average is normally distributed. Moreover, the reasoning provides insight into the properties of the normal distribution and the nature of i.i.d. normal variables.

Introduction

Understanding the distribution of sums or averages of random variables is fundamental in probability and statistics. The CLT plays a crucial role in approximating the distribution of sums of a large number of independent, identically distributed variables that are not necessarily normally distributed, converging towards a normal distribution as the sample size increases. However, when the constituent variables are already normally distributed, the application of the CLT becomes redundant. This paper discusses why this is the case, focusing on the properties of normal random variables and the implications for statistical inference.

Why the CLT is Not Necessary for Normal Variables

The CLT states that the sum (or average) of a large number of independent and identically distributed random variables tends toward a normal distribution, regardless of the original distribution, provided certain conditions (e.g., finite mean and variance) hold. This theorem is particularly useful when dealing with variables that are not normally distributed, enabling practitioners to approximate their sum's distribution with a normal distribution for large samples.

However, if the individual random variables X_i are already normally distributed, such as N(μ₁, σ²₁), then their sum or any linear combination is also normally distributed due to the properties of the normal distribution itself. Specifically, the sum of independent normal variables results in another normal variable whose mean is the sum of the individual means and whose variance is the sum of the individual variances.

This property is an inherent characteristic of the normal distribution, meaning that the normality of variables is preserved under addition, without relying on the CLT. Therefore, in this context, the normal distribution of the sum or average is a direct consequence of the underlying distribution of the individual variables, making the CLT unnecessary.

Mathematical Explanation

Suppose X₁, ..., X_n are independent random variables, each with distribution N(μ₁, σ²₁). The sum S_n = X₁ + ... + X_n then has the distribution:

S_n ~ N(nμ₁, nσ²₁)

This property holds regardless of the size of n. The normality of S_n is intrinsic, as it derives directly from the properties of the normal distribution and the independence of the variables, without requiring the CLT.

Implications for Statistical Practice

This understanding simplifies certain statistical procedures involving normal variables. For example, when summing or averaging normally distributed data, analysts do not need to verify large sample properties or invoke asymptotic theorems to justify normal approximations. Instead, they can directly leverage the distributional properties of the normal distribution for inference, confidence intervals, hypothesis testing, and other statistical analyses.

Conclusion

The key point is that the sum of i.i.d. normal random variables remains normally distributed, making the application of the CLT unnecessary in such cases. This highlights the unique and advantageous properties of the normal distribution, contributing to its prominence in statistical modeling and inference. Recognizing when the CLT applies versus when direct properties suffice is a fundamental aspect of statistical reasoning and effective analysis.

References

• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
• Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer.
• Diaconis, P., & Freedman, D. (1984). Eigenvalues of random matrices: the Bernoulli ensemble. Annals of Probability, 12(2), 376–412.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications (3rd ed.). Wiley.
• Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
• Rice, J. A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Cengage Learning.
• Ross, S. M. (2014). Introduction to Probability and Statistics (5th ed.). Academic Press.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Hogg, R. V., McKean, J., & Craig, A. (2013). Introduction to Mathematical Statistics. Pearson.
• Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.