# Independence and the normal distribution

You know those times that the professor assigns a problem, not to test your knowledge of math, but to test your deep understanding of the material?

Well that's what I'm dealing with right now.

...and I dont get it.

Could anyone take a look at it and maybe give me a hint?

Suppose $X_1, X_2, X_3$ are three iid random variables. We are also told that $X_1 + X_2 + X_3$ and $(X_1 - X_2, X_1 - X_3)$ are independently distributed. Show the common pdf of $X_1, X_2, X_3$ must be the normal pdf

I'm pretty confident it has something to do with the sample mean and sample variance being normally distributed.

The only progress I've made so far is that $X_1 + X_2 + X_3$ is the sample mean, multiplied by n.

• Please add the self-study tag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. – Glen_b -Reinstate Monica Nov 28 '15 at 23:47
• The thread on characterizing normal distributions may provide some inspiration and insight. – whuber Nov 29 '15 at 18:28