I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$.
1) Calculate the density functions, distribution function and characteristic function of $C = \sum_{k=1}^6 C_i$
2) Calculate the probability $P(C > 6100)$
1)
Each $C_i$ is a variable of a normal distribution, so $C$ is a linear combination of these variables therefore $C \sim N(6\cdot1000,6\cdot400) = N(6000,2400)$ So the density and distribution are the ones of a normal distribution. The characteristic function is $\varphi_C(w) = e^{iw6000 -\frac{1}{2} 2400 w^2}$
2)
\begin{align} P(C > 6100) & = 1-P(C \le 6100) = 1 - P\left( \frac{C - 6000}{\sqrt{2400}} \le \frac{6100-6000}{\sqrt{2400}}\right) \\[6pt] & = 1- \Phi\left(\frac{6100-6000}{\sqrt{2400}}\right) \end{align}
Where $\Phi$ is the standard normal distribution function.
I would like to know if it is solved in the right way.