Let $X_i$ be iid normal random variables with mean 0 and standard deviation $\sigma$. Is there a straightforward formula to compute the conditional probability $\mathbb{P}(\sum_{i=1}^{k}X_i < a\:\vert\: \sum_{i=1}^{k-1}X_i < a)$?
If someone can give me a hint, that would be great. Thanks in advance.
Best Answer
Let $S_k = \sum\limits_{i=1}^k X_i$ where $\{X_i\}\mathop{\sim}^\textsf{iid}\mathcal{N}(0,1^2)$
Since $\{X_i\}$ are iid, then $X_k$ and $S_{k-1}$ are independent.