### Distribution of segments on Gaussian chain

The probability distribution function of $P_i(\mathbf{r}_i-\mathbf{r}_\mathrm{cm})$ of ideal chain is evaluated, for $P_i$ represents the probability distribution fucntion of $i$th segment with respect to its centre of mass on the ideal chain. An ideal chain is modeled as multidimensional random walk, where the steps are independent, and the distribution of a step at mean length $b$ is given by $P(\mathbf{r})\sim\mathcal{N}(0,b^2)$. Let $\mathbf{b}_i=\mathbf{r}_{i+1}-\mathbf{r}_{i}$ be the $i$th bond vector, then we have $$\mathbf{r}_i=\sum_{j=1}^{i-1} \mathbf{b}_j$$ and the centre of mass $\mathbf{r}_\mathrm{cm}=\frac{1}{N}\sum_i \mathbf{r}_i$ is $$\mathbf{r}_\mathrm{cm}=\frac{1}{N}\sum_{j=1}^{N-1}(N-j)\mathbf{b}_j$$ then $$\mathbf{r}_i-\mathbf{r}_\mathrm{cm}=\sum_{j=1}^{i-1}\frac{j}{N}\mathbf{b}_j+\sum_{j=i}^{N-1}\frac{N-j}{N}\mathbf{b}_j$$ If variable $X$ is a Gaussian random variable with scale of $\sigma^2$, then $aX$ is a Gaussian random variable with scale of $a^2\si