Probability of having a first occurence in Poisson random distribution

Question

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

Answer 1

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.

$$\mathsf P(X_{1}=0)=???\qquad:t_1=7500$$