Question: Let X be the number of purchases that Fred will make on the online site for a certain company (in some specified time period). Suppose that the PMF of X is P(X = k) = e^-λ/k! for k = 0, 1, 2,... . This distribution is called the Poisson distribution with parameter , and it will be studied extensively in later chapters.

(a) Find P(X ≥ 1) and P(X ≥ 2) without summing infinite series.

(b) Suppose that the company only knows about people who have made at least one purchase on their site (a user sets up an account to make a purchase, but someone who has never made a purchase there doesn't appear in the customer database). If the company computes the number of purchases for everyone in their database, then these data are draws from the conditional distribution of the number of purchases, given that at least one purchase is made. Find the conditional PMF of X given X ≥ 1. (This conditional distribution is called a truncated Poisson distribution.)