Computer scientists have modeled the length of search queries on the web using a Poisson distribution. Suppose the average search query contains about three words. Let X be the number of words in a search query. Since one cannot have a query consisting of zero words, we model X as a €œrestricted€ Poisson distribution that does not take values of 0. That is, let P(X = k) = P(Y = k|Y ‰  0), where Y ˆ¼ Pois(3).

(a) Find the probability function of X.
(b) What is the probability of obtaining search queries longer than 10 words?
(c) Arampatzis and Kamps (2008) observe that many data sets contain very large queries that are not predicted by a Poisson model, such as queries of 10 words or more. They propose a restricted Poisson model for short queries, for instance queries of six words or less. Find P(Y = k|1 ‰¤ Y ‰¤ 6) for k = 0, . . . , 6.

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