Question: If the lifespan of resistors is normally distributed with a mean of $\mu$ years and standard deviation of $\sigma$, what is the expected lifetime a of circuit with $N$ resistors in series (all resistors must be functional in order for the circuit to work)?
My attempts: I feel like it may be the mean, but I also feel like that implies that there are more scores above the mean, which cannot be true (so it has to be less than the mean). I suspect it may also be the first quartile although I haven't proven that this is the case, let alone do I know how I would. Is there a way to solve these sorts of problems? It isn't from a textbook or anything, it's just a question I'm curious about.
Any guidance or help would be appreciated!
Let $X_1,...,X_N$ be independent random variables drawn from the $\mathcal{N}\left(\mu,\sigma^2\right)$ distribution. What you want is $\mathbb{E}\left(\min\left\{X_1,...,X_N\right\}\right)$, because the system breaks as soon as one of the components breaks. The quantity $\min\left\{X_1,...,X_N\right\}$ is itself a random variable, whose distribution is not the same as that of the individual lifetimes, and its mean is difficult to compute when the individual lifetimes are normally distributed.
Are you sure that the individual resistor lifetimes should be normal? Usually in these "reliability" problems the distribution of choice is exponential. In that case the minimum is much more tractable (you can show that it is also exponentially distributed), and also the normal distribution does not make sense for modeling lifetime since it can take on negative values.
For the exponential case, if $X_i$ follows an exponential distribution with parameter $\lambda_i$, then $\min\left\{X_1,...,X_N\right\}$ is exponential with parameter $\sum_i \lambda_i$. The calculation of the mean is then easy.