An example showing that the sum of two normal random variables may not be normal

Abstract

Two novel proofs show that the sum of a specific pair of normal random variables is not normal are established in this note. This is one of the most often misunderstood facts by first-year students in probability theory and statistics. The first proof is concise using the moment generating function. The second proof checks whether the moments of the sum have the property of normal distribution.

Keywords:

• Probability
• normal distributions
• counterexamples
• moment generating functions
• gamma functions

2020 Mathematics Subject Classifications:

• 60E05
• 62E10
• 60E10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI [grant number 22K13433].