In an engine factory the time of work of machines is a random variable $X$. The random variable has an exponential distribution with parameter $\lambda = \frac{1}{1000}$.
It is known that the machine has already been working for $100$ hours. What is the probability that it will be able to work for another $500$ hours?
How do I solve it? My attempt:
$P(X>500+100 | X = 100) = P(X > 600 | X = 100) = P(X > 500)$
This is how I started, by applying memorylessness.
Continuing:
$$P(X > 500) = \int_{500}^{\infty}\frac{1}{\lambda}e^{-\frac{x}{\lambda}}dx$$
$$P(X > 500) = \int_{500}^{\infty}\frac{1}{\frac{1}{1000}} e^{-\frac{x}{\frac{1}{1000}}}dx$$
$$P(X > 500) = \int_{500}^{\infty} 1000 e^{-1000x}dx = \underbrace{\color{red}{\frac{1}{e^{500000}}}}_{\text{nonsense?}}$$
On the other hand, if I do this:
$$P(X > 500) = \int_{500}^{\infty} \frac{1}{1000} e^{-\frac{x}{1000}}dx \approx 0.60653$$ which seems like a more legitimate value.
Would anyone help me formalize my calculations? What was wrong with my formulas? Also, perhaps I should have used $\geq$ symbol instead of $>$ somewhere.
Your "nonsense" answer is the result of using $\lambda$ as though assuming that $\lambda = \frac1{1000}$ means that the machine is expected to break after $\frac1{1000}$ hours. Clearly, the probability of it lasting for $500$ hours is quite low in that case.
Whether $\lambda$ represents the expected time until the first event, or the expected frequency of events, is a matter of convention. Which convention you follow will affect whether you should use $\lambda x$ or $\frac x\lambda$ in your integrals.
For continuous random variables, there is little difference between $>$ and $\geq$. So you can mostly use whichever you like.