Questions: Normal Distribution: Properties and Fundamentals

Question 1

X ~ Normal(0, 9) and Y ~ Normal(0, 16) are independent. What is the distribution of X + Y?

• A. Normal(0, 25) — variances add: 9 + 16 = 25
• B. Normal(0, 49) — standard deviations add: σ_X = 3, σ_Y = 4, so σ = 7 and σ² = 49
• C. Normal(0, 5) — the sum has standard deviation equal to the larger minus the smaller
• D. The sum is not normally distributed — only the original variables are normal

Answer: (A)

For independent normal random variables, variances add: Var(X + Y) = 9 + 16 = 25. So X + Y ~ Normal(0, 25), with σ = 5. The tempting error is adding standard deviations: σ_X = 3, σ_Y = 4, giving σ = 7 — but standard deviations do NOT add. Variances do. This is a critical practical distinction: if you model two independent error sources as Normal(0, σ₁²) and Normal(0, σ₂²), the combined error has variance σ₁² + σ₂², not standard deviation σ₁ + σ₂.

Question 2

Why does the normal distribution appear so frequently in natural measurements like human heights, measurement errors, and test scores?

• A. Because nature produces symmetric distributions, and symmetry implies normality
• B. Because statisticians prefer the normal distribution and routinely fit it to data regardless of its actual shape
• C. Because measurements that arise as the sum of many small independent contributions inevitably approach the normal distribution
• D. Because the normal distribution is the simplest possible continuous distribution and serves as the default assumption

Answer: (C)

This is the content of the Central Limit Theorem, which you'll prove formally in a subsequent topic. Human height is determined by hundreds of genetic and environmental factors; measurement error is the sum of many small instrument and observer variations; test scores aggregate many individual question performances. In each case, the measured quantity is a sum of many independent contributions. The CLT guarantees convergence to normality regardless of the shape of each individual contribution — which is why the normal appears everywhere additive processes operate.

Question 3

Any normal random variable X ~ Normal(μ, σ²) can be converted to a standard normal Z ~ Normal(0, 1) by the transformation Z = (X − μ)/σ.

• T. True
• F. False

Answer: True

Subtracting μ shifts the distribution so its mean is 0; dividing by σ scales it so its standard deviation is 1. This standardization works for any normal, which is why probability tables and software only need to tabulate one distribution — the standard normal. Every probability calculation about any normal distribution reduces to a computation about Z ~ Normal(0, 1).

Question 4

If X ~ Normal(μ₁, σ₁²) and Y ~ Normal(μ₂, σ₂²) are independent, then the standard deviation of X + Y equals σ₁ + σ₂.

• T. True
• F. False

Answer: False

The variance of X + Y equals σ₁² + σ₂² — variances add, not standard deviations. The standard deviation of X + Y is therefore √(σ₁² + σ₂²), not σ₁ + σ₂. These are equal only if one standard deviation is 0. This distinction matters whenever you're aggregating independent sources of uncertainty: adding σ overestimates the combined spread.

Question 5

Explain why the closure property of the normal distribution under linear combinations requires variances — not standard deviations — to add. Why does this distinction matter in practice?

Model answer: Variance is the quantity that adds for independent random variables because of how expectation operates on squared deviations: Var(X + Y) = Var(X) + Var(Y) when X and Y are independent. Standard deviation is the square root of variance, and √(a² + b²) ≠ a + b in general (by the triangle inequality). In practice, this means modeling combined uncertainty correctly requires working with variances, then taking the square root at the end — not adding the individual standard deviations first.

The additivity of variance for independent variables is a consequence of the linearity of expectation applied to squared deviations. It does not generalize to standard deviations. The practical error this prevents: if two independent measurement instruments each have standard deviation 2, the combined measurement error has standard deviation √(4 + 4) = √8 ≈ 2.83, not 2 + 2 = 4. Adding standard deviations overestimates combined uncertainty by assuming perfect positive correlation.