I'm interested in finding the distribution of $\sum_{i=1}^n iX_i$, where $X_i \sim N(i\mu,1)$ independent, as part of the problem that requires to find the MP level $\alpha$ test for $H_0 \, \mu = \mu_0$ vs. $H_1 \, \mu = \mu_1$, where $\mu_1>\mu_0$.
$\textbf{My attempt}$ edited after reading the comments below:
Since $X_i \sim N(i\mu, 1)$ and the Normal family is a scale family, we have that $iX_i \sim N(i^2\mu, i^2)$. This implies that $\sum_i iX_i \sim N(\sum_i i^2\mu, \sum_i i^2)$.
Thank you for your help!
I'm confused. If $$X_i \sim \operatorname{Normal}(i \mu, \sigma^2 = 1),$$ then $$iX_i \sim \operatorname{Normal}(i^2 \mu, \sigma^2 = i^2).$$ That is to say, both the mean and the variance scale, and not in the way you are suggesting. Then $$\sum_{i=1}^n i X_i \sim \operatorname{Normal}(\mu^*, \varsigma^2),$$ where $$\mu^* = \mu \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \mu,$$ and $$\varsigma^2 = \frac{n(n+1)(2n+1)}{6}.$$