Three independently selected objects and one object counted three times are different random models. Read what is being repeated before choosing the variance.
Situation
Random variable
Variance
Three independent observations, each with variance \(\sigma^2\)
\(X_1+X_2+X_3\)
\(3\sigma^2\)
Three times one observed value
\(3X\)
\(9\sigma^2\)
Both have mean \(3\mu\). If the individual observations are normal and independent, their sum is normal. Multiplying a single normal variable by 3 also preserves normality, but scales its variance by \(3^2\).
Bridge the idea
A package contains three independently selected bottles. Compare its total liquid volume with a machine reading that is three times the volume of one bottle. The package uses three random observations; the reading repeatedly uses the same observation.
Write the variables first: \(T=X_1+X_2+X_3\) versus \(R=3X\). Then state independence only for the distinct bottle volumes when needed. You cannot make three copies of \(X\) independent by assigning them different labels.
Exam wording: “The volumes of the three independently selected bottles are assumed independent, so the total variance is the sum of their individual variances.”

Three independently selected objects and one object counted three times are different random models. Read what is being repeated before choosing the variance.

Three independent observations, each with variance \(\sigma^2\)

Both have mean \(3\mu\). If the individual observations are normal and independent, their sum is normal. Multiplying a single normal variable by 3 also preserves normality, but scales its variance by \(3^2\).

A package contains three independently selected bottles. Compare its total liquid volume with a machine reading that is three times the volume of one bottle. The package uses three random observations; the reading repeatedly uses the same observation.

Write the variables first: \(T=X_1+X_2+X_3\) versus \(R=3X\). Then state independence only for the distinct bottle volumes when needed. You cannot make three copies of \(X\) independent by assigning them different labels.

Exam wording: “The volumes of the three independently selected bottles are assumed independent, so the total variance is the sum of their individual variances.”

Repeated Observations and Scaling One Observation

Repeated Observations and Scaling One Observation

Situation	Random variable	Variance
Three independent observations, each with variance \(\sigma^2\)	\(X_1+X_2+X_3\)	\(3\sigma^2\)
Three times one observed value	\(3X\)	\(9\sigma^2\)