Hei,
I was trying to calculate the expected value of Treatment sum of squares and I saw this ...
$$E(\text{SSTr})=(I-1)\sigma^2+J\left(\sum_{I}{(\mu_i^2-\mu^2)}\right) = (I-1) \sigma^2+J\left(\sum_I (\mu_i-\mu)\right)=$$
My question is is this correct and how to prove that(I tried to come from righ to left but it didnt work out)
$$\sum_I (\mu_i^2-\mu^2) = \sum_I(\mu_i-\mu)^2$$
$I$ : is number of treatments
$\mu$: is the total expected value
$\mu_i$: the expected value of each treatment
$J$: number of observations
$$\sum_{i=1}^{n}(\mu_i-\mu)^2=\sum_{i=1}^{n}(\mu_i^2-2\mu_i\mu+\mu^2)=\sum_{i=1}^{n}(\mu_i^2+\mu^2)-2\mu\sum_{i=i}^{n}\mu_i=\sum_{i=1}^{n}(\mu_i^2+\mu^2)-2n\mu*\mu$$
$$\sum_{i=1}^{n=1}(\mu_i^2+\mu^2)-2n\mu^2=\sum_{i=1}^{n}(\mu_i^2+\mu^2) -\sum_{i=1}^{n}2\mu^2=\sum_{i=1}^{n}(\mu_i^2+\mu^2-2\mu^2)=\sum_{i=1}^{n}(\mu_i^2-\mu^2)$$
$$\mathcal{Q.E.D}$$
Hope this is helpful.