# Thread: A simple identity to verify

1. ## A simple identity to verify

Hello,

I am having a hard time with this problem (picture attached). I always have difficulty with proof-like problems such as this one. I guess I just don't really have the mathematical intuition needed for these types of problems.

I am not seeing how this identity works, because I thought my professor said that mew is equivalent to x bar, so I don't even understand how this identity makes sense. Can somebody help me with this problem?? I am supposed to verify the identity shown in the picture.

2. Hi

$\sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 \bar{X} \sum_{i=1}^n X_i + n \bar{X}^2 + n \bar{X}^2 - 2 n \mu \bar{X} + n \mu^2$

$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$

therefore $- 2 \bar{X} \sum_{i=1}^n X_i = - 2 n \bar{X}^2$ cancels out with $n \bar{X}^2 + n \bar{X}^2$

$\sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 n \mu \bar{X} + n \mu^2$

$\sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 \mu \sum_{i=1}^n X_i + n \mu^2$

$\sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i - \mu \right)^2$

3. Thanks for the help!