Let $X_1,X_2,…$ be independent with the common normal density $\eta$, and $S_k= X_1+⋯+X_k$. If $m
find the joint density of $(S_m,S_n)$ and the conditional density for $S_m$ given that $S_n=t$.
Solution attempt
Let the independent random variables have $X_1,X_2,…$ with the common normal density $\eta$ have parameters $(\mu,\sigma^2)$. Then both $S_m$ and $S_n$ have normal densities with parameters$(m\mu,m\sigma^2)$ and $(n\mu,n\sigma^2)$ respectively and there densities are $f_{S_m}$ and $f_{S_n}$.
We also observe that $S_m$ and $S_{n-m}$ are two independent random variables with density $f_{S_m}$and $f_{S_{n-m}}$ and the density of their sum $S_n$ is $f_{S_n}$. The pairs $(S_m,S_n )$ and $(S_m,S_{n-m})$ are related by linear transformation $S_m= S_m,S_{n-m}=S_n-S_m$ with determinant 1. So the joint density of $(S_m,S_n)$ is given by $f_{S_m}(x) f_{S_{n-m}}(s-x)$
$f_{S_m} (x) f_{S_{n-m}} (s-x)=\dfrac{1}{\sqrt {2\pi m \sigma ^{2}}}e^{-\dfrac {\left( x-m \mu\right) ^{2}} {2m \sigma ^{2} }}\dfrac{1}{\sqrt {2\pi (n-m) \sigma ^{2}}}e^{-\dfrac {\left( s-x-(n-m)\mu\right) ^{2}} {2(n-m) \sigma ^{2} }}$
$f_{S_m} (x) f_{S_{n-m}} (s-x)=\dfrac{1}{2\pi\sigma ^{2}\sqrt { m(n-m) }}e^{\dfrac {-(n-m)\left( x-m \mu\right) ^{2}-m\left( s-x-(n-m)\mu\right) ^{2}} {2m(n-m) \sigma ^{2} }}$
I can 't seem to be able to put this into the standard bi-variate normal distribution form.
This appears like a bivariate normal density which tallies with the answer provided by the author but he also states a bivariate normal density with variances $m, n$ and covariance $\sqrt {\dfrac {m} {n}}$.