Show that for a Poisson random variable Y, 2 = . ( Use the result of

Show that for a Poisson random variable Y, σ² = λ. ( Use the result of Exercise 4.83 and Theorem 4.4.)

Data from Exercise 4.83

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Elevator passenger arrivals. A study of the arrival process of people using elevators at a multi-level office building was conducted and the results reported in Build- ing Services Engineering Research and Technology (Oct., 2012). Suppose that at one particular time of day, elevator passengers arrive in batches of size 1 or 2 (i.e., either 1 or 2 people arriving at the same time to use the elevator). The researchers assumed that the number of batches, N, arriving over a specific time period follows a Poisson process with mean A = 1.1. Now let XN represent the number of passengers (either 1 or 2) in batch N and as- sume the batch size has a Bernoulli distribution with p = P(XN = 1) = .4 and q = P(XN = 2) = .6. Then, the total number of passengers arriving over a specific N E-1=X;. The researchers showed that time period is Y = if X1, X2, ..., Xy are independent and identically distrib- uted random variables and also independent of N, then Y follows a compound Poisson distribution. a. Find P(Y = 0), i.e., the probability of no arrivals during the time period. b. Find P(Y = 1), i.e., the probability of only 1 arrival during the time period. c. Find P(Y = 2), i.e., the probability of 2 arrivals during the time period. d. Find P(Y = 3), i.e., the probability of 3 arrivals during the time period.