Determine if the following statements are true, false, uncertain, where $\bar z$ is the simple average of $z_i$ and state any necessary assumptions:
a) $\sum_{i=1}^N X_i\bar Y = \bar X \bar Y$
b)$\sum_{i=1}^N X_iY_i = \bar X \bar Y$
c)$\sum_{i=1}^N X_i^2 = \bar X^2$
d)$\sum_{i=1}^N \sum_{j=1,j\neq i}^N X_iY_j = (X_1\sum_{j=1,j\neq i}^N Y_j + X_2\sum_{j=1,j\neq i}^NY_j +...+X_N\sum_{j=1,j\neq i}^NY_j)$
I figured a) I believe: $$\sum_{i=1}^N X_i\bar Y =\bar Y \left(\frac NN \right) \sum_{i=1}^N X_i = \bar X \bar Y$$ and I am also sure that b) is false, since you cannot expand multiplication like this.
Could you please help to deal with the following question? Thank you!
For a) $$\sum_{i=1}^N X_i\bar Y =\bar Y \sum_{i=1}^N X_i= \bar Y (N\bar X)\not= \bar X \bar Y.$$ If you have forgot a factor $\frac{1}{N}$, then it would be true.
b) is false both since the factor $N^{-2}$ is missing and since you cannot expand like that, as you say (all the cross-terms are missing).
c) is false, since the variance per definition would otherwise always be zero (again, a factor $\frac{1}{N}$ is probably missing).
Regarding d), what do you think? Let me know in a comment and I'll tell you if it is right or wrong.
And good job figuring out the MathJax!