For independent random variables $Y_1,...,Y_n$ with distribution $N(0,1)$ I've been told to prove that $$\overline{Y}, Y_1-\overline{Y}, Y_2-\overline{Y}, ..., Y_n - \overline{Y}$$ are independent by showing that the moment generating function (call it $\varphi(t,s_1,s_2,...,s_n)$) of $$(\overline{Y}, Y_1-\overline{Y}, Y_2-\overline{Y}, ..., Y_n - \overline{Y})$$ factors into $$MGF_{\overline{Y}}(t)\cdot MGF_{Y_1-\overline{Y}}(s_1)\cdot MGF_{ Y_2-\overline{Y}}(s_2) ...MGF_{ Y_n - \overline{Y}}(s_n).$$ Call the latter function $\psi(t,s_1,s_2,...,s_n)$. I've done these computations a number of times, and every time I get
$$\psi(t,s_1,s_2,...,s_n) = \exp \big[ \frac{1}{2n}(t^2+(n-1)\sum_{i=1}^ns_i^2)\big]$$ $$\varphi(t,s_1,s_2,...,s_n)= \exp \big[ \frac{1}{2n}(t^2+n\sum_{i=1}^ns_i^2- (\sum_{i=1}^ns_i)^2)\big].$$ Which is close but not equal. What am I doing wrong?
Here are my computations:
$$\varphi(t,s_1,s_2,...,s_n) = E[\exp (t\overline{Y}+\sum_{i=1}^ns_i(Y_i-\overline{Y}))]=E[\exp (\sum_{i=1}^n(\frac{t}{n}+s_i-\frac{1}{n}\sum_{j=1}^ns_j)Y_i)]=$$
$$\Pi_{i=1}^nE[\exp ((\frac{t}{n}+s_i-\frac{1}{n}\sum_{j=1}^ns_j)Y_i)]= \Pi_{i=1}^nMGF_{Y_i}(\frac{t}{n}+s_i-\frac{1}{n}\sum_{j=1}^ns_j)=$$ $$\Pi_{i=1}^n \exp\big(\frac{1}{2}(\frac{t}{n}+s_i-\frac{1}{n}\sum_{j=1}^ns_j)^2\big)=$$
$$ \exp\bigg[\frac{1}{2}\sum_{i=1}^n\big(\frac{t^2}{n^2}+s_i^2+\frac{1}{n^2}(\sum_{j=1}^ns_j)^2+2\frac{ts_i}{n}-2\frac{t}{n^2}\sum_{j=1}^ns_j-2\frac{s_i}{n}\sum_{j=1}^ns_j\big)\bigg]= $$
$$\exp\bigg[\frac{1}{2}\big(\frac{t^2}{n}+\sum_{i=1}^ns_i^2+\frac{1}{n}(\sum_{j=1}^ns_j)^2+2\frac{t}{n}\sum_{i=1}^ns_i-2\frac{t}{n}\sum_{j=1}^ns_j-2\frac{1}{n}(\sum_{j=1}^ns_j)^2\big)\bigg]=$$
$$\exp\bigg[\frac{1}{2}\big(\frac{t^2}{n}+\sum_{i=1}^ns_i^2-\frac{1}{n}(\sum_{j=1}^ns_j)^2\big)\bigg].$$
$$\psi(t,s_1,s_2,...,s_n)= MGF_{\overline{Y}}(t)\cdot MGF_{Y_1-\overline{Y}}(s_1)\cdot MGF_{ Y_2-\overline{Y}}(s_2) ...MGF_{ Y_n - \overline{Y}}(s_n)=$$
$$ \exp[\frac{1}{2n}t^2]\Pi_{i=1}^n \exp[\frac{1}{2}\frac{n-1}{n}s_i^2]= \exp\bigg[\frac{1}{2}[\frac{1}{n}t^2+\frac{n-1}{n}\sum_{i=1}^ns_i^2]\bigg] $$ where I used the fact that $Y_i - \overline{Y}$ is normally distributed with mean 0 and variance $\frac{n-1}{n}$ and that a variable $X\sim N(0,\sigma^2)$ has $$MGF_X(s) =\exp[\frac{1}{2}\sigma^2s^2].$$