Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is is true to say that k-dimensional Normal distribution is equivalent to the multiplication of k 1-dimensional Normal distributions if variance is equal in all dimensions?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

No. That's true only if all of the covariances are zero. One of the simplest counterexamples is this: suppose $X \sim N(0,1)$. Then $(X,X)$ has a bivariate normal distribution in which the two components are perfectly correlated. The covariance matrix in this case is singular.

Another simple case is this: Suppose $X \sim N(0,1)$ and $Y\sim N(0,1)$ and $X,Y$ are independent, so that the pair $(X,Y)$ has just the sort of distribution that your question anticipates. Then the pair $(X,X+Y)$ has a bivariate normal distribution with correlated components that is not of the kind that your question contemplates. The matrix of covariances is $\begin{bmatrix} 1 & 1 \\ 1 & 2\end{bmatrix}$. The correlation between the two components is $1/\sqrt{2}$, which is about $0.7$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.