**Title: Poisson Distribution and Earthquake Probability Analysis**---**Introduction:**This exercise explores the use of the Poisson distribution to model the occurrence of earthquakes in a specific area. Given an average rate of 2 earthquakes per year, we will calculate various probabilities. ---**Problem (a):***Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area.***Solution Approach:**1. Identify the lambda (λ) for one year, which is 2.2. Use the Poisson probability formula: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]3. Compute the probabilities for k = 1, 2, and 3.4. Sum the probabilities: \( P(1) + P(2) + P(3) \).---**Problem (b):***Find the probability that there will be exactly 5 earthquakes during the next 3-year-period.***Solution Approach:**1. For a 3-year period, the average rate (λ) becomes 2 * 3 = 6.2. Use the Poisson probability formula for k = 5: \[ P(X = 5) = \frac{e^{-6} \times 6^5}{5!} \]---**Problem (c):***Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?***Solution Approach:**1. Calculate the probability of 0 earthquakes in a year: \( P(0) \) using λ = 2: \[ P(X = 0) = \frac{e^{-2} \times 2^0}{0!} \]2. Use the binomial distribution to find the probability of at least 3 years with 0 earthquakes out of 10 years.3. Calculate for k = 3 to 10 using the formula: \[ P(Y = k) = \binom{10}{k} (P(0))^k (1 - P(0))^{10-k} \]4. Sum probabilities for k = 3 to 10.---**Conclusion:**This problem set illustrates how the Poisson distribution can

Given that λ =2 Using Poisson distribution

Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an average rate of 2 earthquakes every year. Show your work when solving the following problems. (a) Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area. (b) Find the probability that there will be exactly 5 earthquakes during the next 3-year- period. (c) Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?