# Probability of having a first occurence in Poisson random distribution

In the Poisson process model for events that occur at random in continuous time with a constant rate $$\lambda$$, there are three related probability distributions of results

• the numbers of events occurring in disjoint intervals of length $$t_1, t_2, t_3,...,t_n$$and independent random variables $$X_1, X_2, X_3,...,X_n$$ with $$X_i \sim Poisson(\lambda t_i)$$

• the times between the occurrences of the events are independent continuous random variables $$T_1, T_2, T_3,...,T_n$$ with $$T_i \sim Exponential(\lambda)$$

• the time of the nth event is a continuous random variable $$Y_n$$ with $$Y_n \sim Gamma(n, \lambda)$$ Given that the locations of a sequence pattern in a large genomic segment follow a Poisson process with the rate parameter $$\lambda = 0.001$$

Question: Compute the probability that the first occurrence of the pattern occurs beyond base 7500.

I've been going through it, I want take $$t_1$$ to be 7500 and solve it as a normal distribution with the poisson random variable formula for all values of x. Please advice how to tackle this

• What is "base 7500"? It sounds like you need to compute the probability of zero events in $[0, 7500)$, which you can compute as $e^{-7500\lambda}$. – Aditya Dua Nov 7 '18 at 17:46

You want the probability that $$0$$ is the count of events occuring in the single time interval $$[0;7500)$$; thus that the first event occurs after that interval.
$$\mathsf P(X_{1}=0)=???\qquad:t_1=7500$$