An electronic device contains $k$ identical, independently working sensitive elements whose lifetimes are exponentially distributed exp$(\theta)$. Even if one element does not operate, the entire device rendered out of order and must be sent to a laboratory for repair. $n$ devices were bought and were sent to a laboratory after lifetimes of $t_1,...,t_n$ hours. Based on this data, find the MLEs of $\theta$, the expected lifetimes of a single element and an entire device, and the probability that a device will operate at least a hours.
My process:
Since the elements are identical and independent, I think $t_i$ is exp$(k\theta)$ since each device can complete $k\theta$ amount of work in per unit time. So $L(\theta;t)=\prod (k\theta)e^{-k\theta t}=(k\theta)^ne^{-k\theta\sum t_i}$. Taking log likelihood we get $l(\theta;t)=n\ln(k)+n\ln(\theta)-k\theta\sum t_i$, taking derivative, setting it to $0$, we get $\hat\theta=\frac{n}{k\sum t_i}$.
Then for the expected lifetime for a single element, I am not sure to use estimate$(E(t)=\frac{1}{\hat\theta}=\frac{k\sum t_i}{n})$ or true mean $(E(t)=\frac1\theta)$. Same issue for the entire device expected lifetime.
For $P(t\ge a)$, I have $P(t\ge a)=1-P(t<a)=1-(1-e^{-k\theta a})=e^{-k\theta a}$. Here I also don't know to use estimation or true parameter.
Could someone clarify?
You are right that $t_i\sim \exp(k\theta).$ More formally, $t_i\le T$ if and only if each of the $k$ devices lasts longer than $T.$ Each has a probability $e^{-\theta T}$ of doing so and is independent so $P(t_i\le T) = (e^{-\theta T})^k.$ From there your maximum likelihood estimation for $\theta$ is correct.
For the second part, I'm confused too. When I hear "expected lifetime" I think of the true mean $\frac{1}{\theta},$ which is of course unknown. If you plug in the MLE you get an ML-estimated version of this which based on the question structure is probably what they're looking for but I don't think they phrased it well. You could also interpret what they asked in a Bayesian way (i.e. what is the expected value of the lifetime under the posterior distribution), which would actually make the most sense here, but I doubt that's what they're going for since they haven't given or mentioned priors.
Same goes for the others of course.