Please use R studio if required both mathematical formula and R function

2. A particular crime occurs randomly in a particular region of Auckland. The findings have shown that this type of crime occurs at a rate of approximately 1.3 per day, and that the number of crimes occurring per day is well-modelled by a Poisson random variable. Total for Question 2: 20 marks (a) What is the probability that exactly 1 crime of this type will occur on a given day? (2 marks) Compute the probability in two ways, using: . the relevant mathematical formula and . an R probability function. (b) What is the probability that exactly 400 crimes will occur in a given year (365 days)? (3 marks) Compute the probability in two ways, using: . the relevant mathematical formula and . an R probability function. (c) What is the expected number of crimes of this type that occur in a given year? (3 marks) (d) What is the probability that between 400 and 500 (inclusive)' crimes of this type occur in a given (4 marks) year? Compute using R and include appropriate mathematical notation as part of your answer. (e) Verify your answers to the parts above using simulation with rpois (). Use 103 simulations. (8 marks) Provide a brief explanation comparing your simulated answers with your theoretical ones