A meteorite shower is a poisson arrival with a rate of 16.6 per minute. Given that 7 meteorites were observed during the first minute, what is the expected value of the time passed until the 10'th meteorite is observed
Extracting info from the question gives us: $$N_t \sim Pois(16.6t)$$ $$T_{i+1}-T_i \sim Exp(16.6)$$ $$T_i \sim Gamma(i,16.6)$$
We want to find $E(T_{10}|N_1=7)$. These two variables are dependent so by definition: $$E(T_{10}|N_1=7)=\int_{0}^{\infty} tP(T_{10} = t|N_1=7) dt$$ The only thing i could extract from that conditional probability is that the integral should start from 1, because the tenth arrival could not have been during the first minute. Also, i don't know any good way of opening up the following: $$=\int_{1}^{\infty} t\dfrac{P(T_{10} = t,N_1=7)}{N_1=7)} dt$$
Is there an easier way to approach this?
The question is equivalent to asking how much time, on average, it will take to observe three more meteorites. This is just the expectation of a gamma random variable with shape $n = 3$ and rate $\lambda = 16.6$; i.e., it will take on average $3/16.6 \approx 0.180723$ minutes, or about $10.8$ seconds. This does not include the one minute that it took to observe $7$ meteorites.