Discussion
Suppose \(Y_{1}\), \(Y_{2}\) and \(Y_{3}\) are independent and identically distributed \(p\)-variate random vectors having normal distribution. Then the distribution of \(a_{1}Y_{1} + a_{2}Y_{2} + a_{3}Y_{3}\), where \(a_{1}, a_{2}, a_{3}\) are vectors of order \(1 \times p\), is :
Answer: A
The distribution is univariate normal.
Each \(a_{i}\) is a \(1 \times p\) row vector and each \(Y_{i}\) is a \(p \times 1\) normal vector, so \(a_{i}Y_{i}\) is a scalar, a linear combination of the components of \(Y_{i}\). A linear combination of a multivariate normal vector is univariate normal, and the three terms are independent, so their sum is univariate normal with mean \(\sum a_{i}\mu\) and variance \(\sum a_{i}\Sigma a_{i}^{t}\).
The result is a single number, not a vector, so it cannot be trivariate or four-variate; and linear functions of normal vectors are always normal.
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