Number of meters between $2$ Ant Coloneys in a road Is Exponential random variable with Variance $25 ~(meter)^2$.
in $1$ KiloMeters we counted $100$ Coloneys . Write a Closed Expression in which we can calculate with Computer the probability of counting at least $150$ Aditional colones in that kilometer.
You are forbidden to put the sign $\infty$ in your Final answer.
Answer Attempt :
First Let N be random Vairable with $N$~$Poison$ $(\lambda = \frac{1}{\lambda_{exp}} = 200)$ the number of colonies per kilometer.
$P(N) = \frac{e^{-200}~200^n}{n!}$
$P(100+150\leq N|100<N) = \frac{P(250\leq N)}{P(100<N)} = \frac{1- \int_0^{250}{\frac{e^{-200}~200^n}{n!}dn}}{1- \int_0^{100}{\frac{e^{-200}~200^n}{n!}dn}}$
First is this the correct answer
second how can one continue to get closed expression
third why did they write the last sentence and include computer information
thanks.