In my A level stats exam a question was to show that: $$E(e^{kt})=\frac{1}{1-4k}$$
I took the term $E(e^{kt})$ to mean $E(X)$ where X is crv with a pdf $f(t)$ and $f(t)=e^{kt}, a<x<b$ (and zero otherwise). I then tried to show that:
$$\int_{-\infty}^{\infty}tf(x)dt = \frac{1}{1-4k}$$
Which isnt true. In hindsight its clear that the exam board meant to write $E(e^{kT})$ where $T$ is a crv defined in an earlier part of the question whith a cdf $F(t)=1-e^{-0.25t},t>0$. In this case the question would require showing that:
$$\int_{0}^{\infty}e^{kt}\frac{d}{dt}F(t)dt = \frac{1}{1-4k}$$
My question is: Is my reasoning correct given the notation of the question and would it be resonable to ask for compensation for the time I wasted trying to prove something that wasnt true?